Large Parallel Volumes of Finite

and Compact Sets in d-Dimensional Euclidean Space

Jürgen Kampf and Markus Kiderlen^{1}

Received: January 10, 2012 Revised: June 18, 2012

Communicated by Lars Hesselholt

Abstract. The r-parallel volume V (C_{r}) of a compact subset C in d-dimensional Euclidean space is the volume of the set C_{r} of all points of Euclidean distance at most r > 0 from C. According to Steiner’s formula, V (C_{r}) is a polynomial in r when C is convex. For finite sets C satisfying a certain geometric condition, a Laurent expansion of V (C_{r}) for large r is obtained. The dependence of the coefficients on the geometry of C is explicitly given by so-called intrinsic power volumes of C. In the planar case such an expansion holds for all finite sets C. Finally, when C is a compact set in arbitrary dimension, it is shown that the difference of large r-parallel volumes of C and of its convex hull behaves like cr^{d−3}, where c is an intrinsic power volume of C.

2000 Mathematics Subject Classification: Primary 52A39; Secondary 52B11, 41A10. Keywords and Phrases: Large parallel sets, Laurent expansion of parallel volume, Steiner formula, intrinsic power volume.

1 Introduction

In 1840, Jakob Steiner [10] showed that the r-parallel volume of certain compact convex sets in ℝ^{d} are polynomials in r ≥ 0 when d = 2 or d = 3. The generalization to arbitrary dimensions is now known as the Steiner formula

V_{d}(C_{r}) = ∑_{i=0}^{d} κ_{i}V_{d−i}(C)r^{i},  r ≥ 0.  (1)

1Both authors were supported by the Villum Foundation in the framework of the Centre for Stochastic Geometry and Advanced Bioimaging, CSGB.

Documenta Mathematica 18 (2013) 275–295

Here C is any compact convex subset of ℝ^{d}, C_{r} is the (outer) r-parallel set of C consisting of all points of distance at most r from C, V_{d} is the Lebesgue measure, κ_{i} denotes the volume of the i-dimensional Euclidean unit ball and V_{i}(C) are the intrinsic volumes of C, which are defined by this relation. A possible proof of (1) uses the fact that the intrinsic volumes can be given explicitly in the case where the set C is a convex polytope P:

V_{i}(P) = ∑_{F ∈F_{i}(P)} γ(F, P)V_{i}(F),  (2)

i = 0, …, d. Here F_{i}(P) is the family of i-dimensional faces of P and γ(F, P) is the outer angle of P at F defined in Equation (13) below. There are numerous generalizations of the Steiner formula, for instance local versions [8, Chapter 4] and even Steiner-type results for arbitrary closed sets C; see [3]. Often, Steiner-type formulas are only valid for sufficiently small r, e.g. in the case of sets of positive reach [1]. In the present work we focus on r-parallel volumes for large r and determine a Laurent series expansion or at least its leading coefficients. We discuss mainly the parallel volume of finite sets, which turns out to be already nontrivial, but Theorem 2, below, deals with compact sets C ⊆ ℝ^{d}. A first result for large parallel volumes has been obtained in [4] for compact sets C ⊆ ℝ^{d}, where it is shown that the volume of C_{r} is close to the volume of its convex hull conv C_{r} = (conv C)_{r}. In fact, there is a constant c = c(C) such that

0 ≤ V_{d}(conv C_{r}) − V_{d}(C_{r}) ≤ cr^{d−3}  (3)

for all sufficiently large r. In [4] an example set C was given where this volume difference behaves like r^{d−3}, so the exponent here is best possible. Independently, a weaker version of (3) was shown in [7, Lemma 2], where C was assumed to be an at most two-dimensional subset of the set {0, 1}^{d}, d ≥ 3, and the volume difference was shown to converge to zero faster than r^{d−2}. In [7] this was used to obtain an (incomplete) collection of asymptotic Miles-type formulas for the specific intrinsic volumes of stationary digitized Boolean models of balls. It was the original motivation for the present research to complete and generalize these results. One might even ask for similar asymptotic formulas for the specific intrinsic volumes of digitized standard random sets. (Standard random sets are stationary, a.s. locally polyconvex random sets, satisfying a certain integrability condition; see [9, Definition 9.2.1]). While the Boolean model case only requires (truncated) Laurent expansions for the volume of the set C_{r}, being the Minkowski sum of C with the r-scaled Euclidean unit ball B^{d}, results for general standard random sets would require such expansions for the volume of the Minkowski sum of C with an arbitrary r-scaled polyconvex set. In [6, Corollary 2.2] the first two leading coefficients of such a Laurent expansion are determined and [6, §5] discusses its application to the theory of random sets, including the calculation of the one-sided derivative of the contact distribution function at zero. However, higher order expansions in this general setting appear not to be known.

Other applications of our expansions appear to be in reach. For example, in [4] formula (3) was used to examine the expected value of the parallel volume of Brownian paths, when the time is small, and to relate analytical properties of r ↦ W (rK) to geometric properties of K, where W denotes the Wills functional and K is a fixed compact set.

In the next section, the main results will be stated starting with the definition of functionals of finite subsets of ℝ^{d} capturing all geometric properties that are relevant when considering large parallel volumes. An optimal lower bound for the constant c in (3) will be given in Theorem 2. For finite sets C, we obtain an explicit Laurent expansion in the case d = 2 in Proposition 1 and, under an additional condition on C, in Theorem 5 for all d ≥ 2. Section 3 provides proofs of the main results and discusses in particular the properties of the coefficients in the Laurent expansion.

2 Main results

The r-parallel volumes of one-dimensional compact sets are trivially affine functions for sufficiently large r. We therefore assume d ≥ 2 throughout the following. In the spirit of (2) define

V_{i}^{(m)}(C) = ∑_{F ∈F_{i}(conv C)} γ(F, conv C) ∫_{F} d(C ∩ F, x)^{m−i} dx,  (4)

where i = 0, …, d, m ≥ i, and C is a finite subset of ℝ^{d}. We call the functionals V_{i}^{(m)} the intrinsic power volumes. Here, integration is understood with respect to i-dimensional Lebesgue measure in the affine hull of F, and d(C ∩ F, x) is the smallest Euclidean distance between x and a point in C ∩ F. Due to (2), we have

V_{i}^{(i)}(C) = V_{i}(conv C)

for all i = 0, …, d. The functionals V_{i}^{(m)}, defined on the family of finite subsets of ℝ^{d}, share many properties with usual intrinsic volumes (see Lemma 9) and are in particular independent of the dimension of the embedding space. The functional V_{1}^{(m)}(C), where the sum in (4) extends over all edges of conv C, is given more explicitly in (17). In particular, if C is the set of vertices of a convex polytope,

V_{1}^{(m)}(C) = \frac{1}{m2^{m−1}} ∑_{F ∈F_{1}(conv C)} γ(F, conv C)V_{1}^{m}(F)  (5)

m ≥ 1. In the case of planar finite sets, these (and the classical intrinsic volumes) are the only intrinsic power volumes that occur in a Laurent expansion of large parallel volumes.

Proposition 1. Let C ⊆ ℝ^{2} be a finite set and put K = conv C. Then

V_{2}(K_{r}) − V_{2}(C_{r}) = 2 ∑_{n=1}^{∞} \frac{(2n − 3)!!}{(2n)!!}V_{1}^{(2n+1)}(C)r^{−(2n−1)}

for all sufficiently large r. (The definition of the double factorial n!! is recalled in Equation (12) below.)

Proposition 1 and Corollary 6 follow directly from Theorem 5 below. All other results in this section will be shown in Section 3. The leading coefficient of the Laurent expansion in Proposition 1 depends on C through the intrinsic power volume V_{1}^{(3)}(C). An analogue statement holds when C is only assumed to be compact, and in all dimensions d ≥ 2. To define V_{1}^{(m)}(C), m ≥ 1, also for compact C ⊆ ℝ^{d}, we choose for any x ∈ ℝ^{d} two points p_{x}^{C} and q_{x}^{C} in C with x in the line segment [p_{x}^{C}, q_{x}^{C}] and C ∩ [p_{x}^{C}, q_{x}^{C}] = {p_{x}^{C}, q_{x}^{C}}, whenever such points exist and are unique up to permutation. Otherwise we put p_{x}^{C} = q_{x}^{C} = x. We then define

V_{1}^{(m)}(C) = \frac{1}{κ_{d−1}} ∫_{ℝ^{d}} d(\{p_{x}^{C}, q_{x}^{C}\}, x)^{m−1} C_{1}(conv C, dx),  (6)

where C_{1}(conv C, ·) is the first curvature measure of conv C. If C is finite, then the definitions (4) and (6) coincide as will be shown in Remark 7.

Theorem 2. If K is the convex hull of a compact set C ⊆ ℝ^{d}, then

lim_{r→∞} \frac{V_{d}(K_{r}) − V_{d}(C_{r})}{r^{d−3}} = \frac{ω_{d−1}}{2} V_{1}^{(3)}(C).  (7)

Here ω_{i} is the surface area of the (i − 1)-dimensional unit sphere in ℝ^{i}.

A natural question is whether the speed of convergence in Theorem 2 can be determined: is there an α > 0 such that, for any compact C ⊆ ℝ^{d}, there is a constant c = c(C) with

|\frac{V_{d}(K_{r}) − V_{d}(C_{r})}{r^{d−3}} − \frac{ω_{d−1}}{2} V_{1}^{(3)}(C)| ≤ cr^{−α}

for all sufficiently large r? The following proposition (with f(r) = r^{−α/2}) shows that, already in ℝ^{2}, such a stability result cannot hold.

Proposition 3. Let f : (0, ∞) → (0, ∞) be a continuous bijective map with lim_{r→∞} f(r) = 0. Then there is a compact set C ⊆ ℝ^{2} and a number c > 0 such that

cf(r) ≤ |\frac{V_{2}(K_{r}) − V_{2}(C_{r})}{r^{−1}} − V_{1}^{(3)}(C)|

for all sufficiently large r. As usual, we have put K = conv C.

Documenta Mathematica 18 (2013) 275–295

However, if the class of compact sets is replaced by the smaller class of all finite sets, a stability result with α = 1 can be obtained. (That this speed of convergence is optimal for this class when d ≥ 3 follows from Theorem 5 below.)

Theorem 4. Let C ⊆ ℝ^{d} be finite and put K = conv C. Then there is a constant c = c(C) such that

|\frac{V_{d}(K_{r}) − V_{d}(C_{r})}{r^{d−3}} − \frac{ω_{d−1}}{2} V_{1}^{(3)}(C)| ≤ \frac{c}{r} ,  (8)

for sufficiently large r.

In the remainder of this section, C will always be a finite subset of ℝ^{d}. Under the following condition on the facets (these are the (d − 1)-dimensional faces) of conv C, we can also obtain a Laurent series expansion of infinite order in higher dimensions, generalizing Proposition 1.

Condition (A). For all facets G of the polytope K = conv C and all faces F of G we have

d(C ∩ F, x) = d(C ∩ G, x) for all x ∈ F.  (9)

Clearly (9) holds whenever F is a singleton or F = G. Hence Condition (A) must only be checked for faces F of dimension between 1 and d − 2, and is in particular satisfied for all finite sets C in ℝ^{2}. In ℝ^{3}, only edges F ∈ F_{1}(K) must be considered. In particular, if C is the vertex set of a simplicial polytope in ℝ^{3}, Condition (A) is violated if and only if at least one facet of conv C is a triangle with a strictly obtuse angle.

For instance, Condition (A) is fulfilled if C ⊆ ℝ^{d} is the set of vertices of a rectangular cuboid. If C is the set of the vertices of the standard simplex

S_{d} = {(x_{1}, …, x_{d}) ∈ ℝ^{d} : ∑_{i=1}^{d} x_{i} ≤ 1, x_{1}, …, x_{d} ≥ 0}

in ℝ^{d}, Condition (A) is satisfied if and only if d ∈ {2, 3}. This can be shown as follows. We have already seen that Condition (A) is trivially fulfilled when d = 2, and it also holds for d = 3 as the triangles forming the facets of S_{3} do not contain a strictly obtuse angle. For d ≥ 4 the point x = (1/3, 1/3, 1/3, 0, …, 0) ∈ ℝ^{d} is contained in the face

F = {(y_{1}, y_{2}, y_{3}, 0, …, 0) ∈ ℝ^{d} : ∑_{i=1}^{3} y_{i} = 1, y_{1}, y_{2}, y_{3} ≥ 0}

of the facet G = S_{d−1} × {0}. Hence (9) is violated, as

d(C ∩ F, x) = \sqrt{\frac{2}{3}} > \sqrt{\frac{1}{3}} = ∥x − o∥ = d(C ∩ G, x).

If C ⊆ ℝ^{d} satisfies Condition (A), a Laurent expansion for large parallel volumes with explicit coefficients can be shown.

Theorem 5. Let K be the convex hull of a finite set C ⊆ ℝ^{d} that satisfies Condition (A). Then

V_{d}(K_{r}) − V_{d}(C_{r}) = ∑_{n=3−d}^{∞} a_{n}(C)r^{−n}

for all sufficiently large r with coefficients

a_{n}(C) = ∑_{{i=1, \atop 2|(n+d−i)}}^{min\{d−1,n+d−2\}}(−1)^{(n+d−i+2)/2}\binom{(d − i)/2}{(n + d − i)/2}κ_{d−i}V_{i}^{(n+d)}(C)

which vanish for all even positive n.

As Condition (A) is satisfied for all planar finite sets, Proposition 1 is a direct consequence of Theorem 5. Indeed, in the case when n > 0 is odd and d = 2 the sum in the definition of a_{n}(C) consists only of one summand, namely

a_{n}(C) = (−1)^{(n+3)/2}\binom{1/2}{(n + 1)/2}κ_{1}V_{1}^{(n+2)}(C) = 2\frac{(n − 2)!!}{(n + 1)!!}V_{1}^{(n+2)}(C).

Also for d = 3 the representation in Theorem 5 simplifies considerably. We get similarly as above

a_{0}(C) = (−1)^{2}\binom{1}{1}κ_{2}V_{1}^{(3)}(C) = πV_{1}^{(3)}(C),

and

a_{n}(C) = (−1)^{(n+3)/2}\binom{1/2}{(n + 1)/2}κ_{1}V_{2}^{(n+3)}(C) = 2\frac{(n − 2)!!}{(n + 1)!!}V_{2}^{(n+3)}(C)

for positive odd n.

Corollary 6. Let K be the convex hull of a finite set C ⊆ ℝ^{3} that satisfies Condition (A). Then, for all sufficient large r,

V_{3}(K_{r}) − V_{3}(C_{r}) = πV_{1}^{(3)}(C) + 2 ∑_{n=1}^{∞} \frac{(2n − 3)!!}{(2n)!!}V_{2}^{(2n+2)}(C) r^{−(2n−1)}. (10)

This section is concluded by an example where Corollary 6 is applied to the vertex set

C^{d} = vert([0, 1]^{d}) = ⋃_{F ∈F_{0}([0,1]^{d})} F

of the unit cube [0, 1]^{d} ⊆ ℝ^{d} for d = 3. It will be shown in Subsection 3.6 below, that (10) then becomes

V_{3}([0, 1]_{r}^{3}) − V_{3}(C_{r}^{3}) = \frac{π}{4} + ∑_{n=1}^{∞} b_{n} r^{−(2n−1)}  (11)

with

b_{n} = \frac{6}{(2n + 1)(2n − 1)(n + 1)4^{n}} ∑_{i=0}^{n} \frac{(2i − 1)!!}{i!}.

3 Proofs and auxiliary results

3.1 Notations and the intrinsic power volumes

The following subsection summarizes the notation and discusses basic properties of the intrinsic power volumes. For the reader’s convenience we repeat the most important concepts already named in the previous sections. The Euclidean norm in ℝ^{d} is denoted by ∥ · ∥, the closed Euclidean unit ball by B^{d} and its boundary, the (d − 1)-dimensional unit sphere by S^{d−1}. We write A^{C}, diam A, bd A, and relint A for the complement, the diameter, the topological boundary, and the relative interior of a set A ⊆ ℝ^{d}, respectively. Let d(A, x) = inf_{y∈A} ∥x − y∥ be the distance of x ∈ ℝ^{d} to A ⊆ ℝ^{d}. Let [x, y] be the line segment with endpoints x, y ∈ ℝ^{d}. We denote the m-dimensional Hausdorff measure by H^{m} and use the volume of the Euclidean unit ball κ_{d} = H^{d}(B^{d}) and the surface area of the (d − 1)-dimensional unit sphere ω_{d} = H^{d−1}(S^{d−1}). Often, we will write V_{d} = H^{d} for the usual Lebesgue measure. The double factorial of an integer n ≥ −1 is defined to be

n!! = \begin{cases}n · (n − 2) · · · · · 2, & if n > 0 is even, \\ n · (n − 2) · · · · · 1, & if n > 0 is odd, \end{cases}  (12)

and the usual convention (−1)!! = 0!! = 1. We recall some basic notions from convex geometry; see [8] for details. A convex body is a nonempty compact convex set in ℝ^{d}. For a convex body K let p(K, x) be the metric projection of x ∈ ℝ^{d} onto K, that is the point in K closest to x. Then d(K, x) = ∥x − p(K, x)∥ is the distance between x and K. If x ∉ K,

u(K, x) = \frac{x − p(K, x)}{d(K, x)} ∈ S^{d−1}

is the negative projection direction. A convex subset F of a convex body K is called face of K if for any two points x, y ∈ K with \frac{x+y}{2} ∈ F we have x, y ∈ F. Let F_{i}(K) be the family of all i-dimensional faces of K, i = 0, …, d. We denote the normal cone of K at F ∈ F_{i}(K) by N(K, F) and put n(K, F) = N(K, F)∩S^{d−1}. Then n(K, F) = {u(K, x) ∈ S^{d−1} : x ∉ K, p(K, x) ∈ relint F }. For a polytope K

γ(F, K) = \frac{H^{d−i−1}(n(K, F))}{ω_{d−i}}  (13)

is the exterior angle of K at F ∈ F_{i}(K), i = 0, …, d − 1. For completeness we set γ(K, K) = 1 if K is full-dimensional. We recall that the support measures Θ_{0}(K, ·), …, Θ_{d−1}(K, ·) of K are the measures on ℝ^{d} × S^{d−1} satisfying

H^{d}({x ∈ ℝ^{d} \ K : d(K, x) ≤ r, (p(K, x), u(K, x)) ∈ η})

= ∑_{m=0}^{d−1} \frac{\binom{d−1}{m}}{d − m}r^{d−m}Θ_{m}(K, η)

for all r ≥ 0 and all Borel sets η ⊆ ℝ^{d} × S^{d−1}. More generally, for any measurable function f ≥ 0 on ℝ^{d} we have

∫_{ℝ^{d}\K} f(x) dx = ∑_{m=0}^{d−1} \binom{d − 1}{m} ∫_{ℝ^{d}×S^{d−1}} ∫_{0}^{∞} f(x + su)×

×s^{d−m−1} ds Θ_{m}(K, d(x, u)).  (14)

The support measures are concentrated on the (generalized) normal bundle

N (K) = {(p(K, y), u(K, y)) : y ∉ K} ⊆ (bd K) × S^{d−1}.

Their total mass is

Θ_{m}(K, ℝ^{d} × S^{d−1}) = ω_{d−m}\binom{d − 1}{m}^{−1}V_{m}(K).  (15)

The area measures of K are the projections of the support measures to their second component:

S_{m}(K, ω) = Θ_{m}(K, ℝ^{d} × ω)

where ω is a Borel set in S^{d−1}. The curvature measures are their projections on the first component given by

C_{m}(K, β) = Θ_{m}(K, β × S^{d−1})

for Borel sets β ⊆ ℝ^{d}. If K is a polytope, then

\binom{d − 1}{m}Θ_{m}(K, η) = ∑_{F ∈F_{m}(K)} ∫_{F} ∫_{n(K,F)} 1_{η}(x, u) dH^{d−m−1}(u)dH^{m}(x) (16)

for any Borel set η ⊆ ℝ^{d} × S^{d−1}; see [8, (4.2.2)].

Remark 7. If C is finite, then (16) implies

C_{1}(conv C, ·) = κ_{d−1} ∑_{F ∈F_{1}(conv C)} γ(F, conv C)H^{1}(F ∩ ·),

and thus, (6) coincides with the definition (4) when i = 1.

For a convex body K ⊆ ℝ^{d} we call

F_{K}(u) = {x ∈ K : ⟨x, u⟩ = h_{K}(u)}

the support set of K in direction u ∈ S^{d−1}, where h_{K} : S^{d−1} → ℝ is the support function of K and ⟨·, ·⟩ denotes the Euclidean scalar product.

Lemma 8. Let K ⊆ ℝ^{d} be a convex body and m ∈ {0, …, d − 2}. Set

A_{m} = {u ∈ S^{d−1} : dim F_{K}(u) ≥ m + 1}.

Then S_{m}(K, A_{m}) = 0.

Proof. As a straight-forward generalization of [8, Theorem 2.2.9], we obtain

H^{d−m−1}(A_{m}) = 0.

By [8, Theorem 4.6.5] there is a constant a such that

S_{m}(K, ω) ≤ aH^{d−m−1}(ω)

for each (H^{d−m−1}, d − m − 1)-rectifiable set ω ⊆ S^{d−1}, where we just want to mention that zero sets are always rectifiable and refer to [2, 3.2.14] for a complete definition. So, S_{m}(K, A_{m}) = 0.

We now summarize properties of the intrinsic power volumes as functionals on the family E of finite subsets C of ℝ^{d}. In order to get a simplified expression for V_{1}^{(m)}(C), m ≥ 1, we define the set

F_{1}^{∗}(C) = {[x, y] : x ≠ y, [x, y] ∩ C = {x, y},
there is e ∈ F_{1}(conv C) with [x, y] ⊆ e}

of all refined edges of conv C, where every edge in F_{1}(conv C) is partitioned into line segments such that exactly their endpoints are in C. Note that if C = F_{0}(K) for a convex polytope K, then F_{1}^{∗}(C) = F_{1}(conv C), but in general this does not hold. For C ∈ E and e ∈ F_{1}^{∗}(C) we put N(conv C, e) = N(conv C, ẽ) and γ(e, conv C) = γ(ẽ, conv C), where ẽ is the unique edge of conv C with e ⊆ ẽ.

Lemma 9 (Properties of V_{i}^{(m)}). Let i = 1, …, d, and m ≥ i be given.

(a) V_{i}^{(i)}(C) = V_{i}(conv C) for any C ∈ E.

(b) V_{i}^{(m)} is homogeneous of degree m:

V_{i}^{(m)}(αC) = α^{m}V_{i}^{(m)}(C)

for all α ≥ 0 and all C ∈ E.

(c) V_{i}^{(m)} is motion invariant.

(d) V_{i}^{(m)} is independent of the embedding space: Let C be a finite set in ℝ^{d′} ⊆ ℝ^{d} for some d′ < d. Then V_{i}^{(m)}(C) for C considered as a subset of ℝ^{d′} coincides with V_{i}^{(m)}(C) for the (lower-dimensional) subset C of ℝ^{d}.

(e) Simplified expression for V_{1}^{(m)}: For finite C ⊆ ℝ^{d} we have

V_{1}^{(m)}(C) = \frac{1}{2^{m−1}m} ∑_{F ∈F_{1}^{∗}(conv C)} γ(F, conv C)V_{1}(F)^{m}.  (17)

Thus, V_{i}^{(m)}, as a functional on E, has similar properties as the intrinsic volume on the family of convex bodies. However, in contrast to the latter, V_{i}^{(m)} is in general not a valuation, i.e. it is not additive, and it is not continuous with respect to the Hausdorff metric. The properties (a)-(d) of Lemma 9 hold for arbitrary compact sets C if i = 1.

3.2 Proof of Theorem 2

The proof of Theorem 2 is divided into a sequence of lemmas. Several times, the following analytical lemma will be needed. It can be shown using a Taylor expansion of order two.

Lemma 10. Let n ∈ ℕ, 0 ≤ a ≤ r, and f_{n}(r) = r^{n} − \sqrt{r^{2} − a^{2}}^{n}. Then

\frac{1}{2}a^{2}r^{−1} ≤ f_{n}(r) ≤ \frac{1}{2}a^{2}r^{−1} + \frac{\sqrt{2}}{4}a^{4}r^{−3}, if n = 1, and a ≤ \frac{r}{2},

\frac{3}{2}a^{2}r − \frac{3\sqrt{2}}{8}a^{4}r^{−1} ≤ f_{n}(r) ≤ \frac{3}{2}a^{2}r, if n = 3, and a ≤ \frac{r}{2},

\frac{n}{2}a^{2}r^{n−2} − \frac{n(n − 2)}{8}a^{4}r^{n−4} ≤ f_{n}(r) ≤ \frac{n}{2}a^{2}r^{n−2}, if n ≥ 4.

In short,

|f_{n}(r) − \frac{n}{2}a^{2}r^{n−2}| ≤ c_{n}a^{4}r^{n−4}  (18)

where c_{2} = 0 and c_{n} = n(n − 2)/8 for n ≥ 4. Inequality (18) also holds for n = 1 with c_{1} = \sqrt{2}/4 and for n = 3 with c_{3} = 3\sqrt{2}/8 if 0 ≤ a ≤ r/2. In particular, the inequality f_{n}(r) ≤ na^{2}r^{n−2}, which is elementary for n = 1, holds for all n.

For compact C ⊆ ℝ^{d} we now show that the difference between the parallel volume of K = conv C and the parallel volume of C is approximately

I_{C}(r) = (d − 1) ∫_{ℝ^{d}×S^{d−1}} ∫_{0}^{∞} 1_{K_{r}\C_{r}}(x + su)s^{d−2} ds Θ_{1}(K, d(x, u)).

Lemma 11. Let C ⊆ ℝ^{d} be a compact set and put K = conv C. Then there is a constant c = c(C) with

0 ≤ V_{d}(K_{r}) − V_{d}(C_{r}) − I_{C}(r) ≤ c · r^{d−4}  (19)

for all r ≥ diam C.

Proof. Without loss of generality, we may assume that the diameter of C is positive. If x ∈ K_{r} \ C_{r}, then due to ext K := ⋃_{F ∈F_{0}(K)} F ⊆ C, we have

p(K, x) ∉ ext K,  (20)

and an application of the Pythagorean theorem implies

d(K, x) > \sqrt{r^{2} − (diam C)^{2}},  (21)

whenever r ≥ diam C; see [5, Example 3.3 and Lemma 3.5]. Since r ≥ diam C implies (K_{r} \ C_{r}) ∩ K = ∅ and (20) together with Lemma 8 implies Θ_{0}(K, {(p(K, x), u(K, x)) | x ∈ K_{r} \ C_{r}}) = 0, we get from (14) with f = 1_{K_{r}\C_{r}} that

V_{d}(K_{r}) − V_{d}(C_{r}) = V_{d}(K_{r} \ C_{r})

= ∑_{m=1}^{d−1} \binom{d − 1}{m} ∫_{ℝ^{d}×S^{d−1}} ∫_{0}^{∞} 1_{K_{r}\C_{r}}(x + su)×

× s^{d−m−1} ds Θ_{m}(K, d(x, u)).  (22)

The term on the right hand side of (22) corresponding to m = 1 is I_{C}(r). We now consider the summands of the right hand side of (22) for which m ≥ 2. For d = 2 no such summands exist. Since all these summands are non-negative for d ≥ 3, the left inequality of the assertion is shown. The right inequality follows from the fact that – due to equations (21) and (15) and Lemma 10 – the right hand side of (22), without the summand for m = 1, is bounded from above by

∑_{m=2}^{d−1} \binom{d − 1}{m} ∫_{ℝ^{d}×S^{d−1}} ∫_{√\overline{r^{2}−(diam C)^{2}}}^{r} s^{d−m−1} ds Θ_{m}(K, d(x, u))

= ∑_{m=2}^{d−1} ω_{d−m} \frac{1}{d − m} (r^{d−m} − (r^{2} − (diam C)^{2})^{(d−m)/2}) V_{m}(K)

≤ ∑_{m=2}^{d−1} ω_{d−m}(diam C)^{2}r^{d−m−2}V_{m}(K)

≤ cr^{d−4}, where c = ∑_{m=2}^{d−1} ω_{d−m}(diam C)^{4−m}V_{m}(K).

An upper bound for I_{C}(r) is obtained easily.

Lemma 12. Let C ⊆ ℝ^{d} be a compact set and put K = conv C. If r ≥ 2(diam C), then

I_{C}(r) ≤ \frac{ω_{d−1}}{2} V_{1}^{(3)}(C)r^{d−3}

for d ≥ 3 and

I_{C}(r) ≤ V_{1}^{(3)}(C)r^{−1} + \frac{\sqrt{2}}{4}V_{1}^{(5)}(C)r^{−3}

for d = 2.

Proof. Recall A_{1} := {u ∈ S^{d−1} | dim F_{K}(u) ≥ 2}. For any (x, u) ∈ ((bd K \ C) × A_{1}^{C}) ∩ N (K) the set F_{K}(u) is at most 1-dimensional due to u ∉ A_{1} and at least 1-dimensional due to x ∉ C. Thus we have p_{x}^{C} ≠ x and q_{x}^{C} ≠ x, where p_{x}^{C} and q_{x}^{C} were defined before formula (6). Now

1_{K_{r}\C_{r}}(x + su) = 1_{[p_{x}^{C},q_{x}^{C}]_{r}\C_{r}}(x + su) ≤ 1_{[p_{x}^{C},q_{x}^{C}]_{r}\{p_{x}^{C},q_{x}^{C}}_{r}}(x + su)

holds for all (x, u) ∈ ((bd K)×A_{1}^{C})∩N (K) and hence for Θ_{1}-a.a. (x, u) ∈ N (K) by Lemma 8. Setting a_{x} = d({p_{x}^{C}, q_{x}^{C}}, x), we get

I_{C}(r) ≤ (d − 1) ∫_{ℝ^{d}×S^{d−1}} ∫_{0}^{∞} 1_{[p_{x}^{C},q_{x}^{C}]_{r}\{p_{x}^{C},q_{x}^{C}}_{r}}(x + su) s^{d−2} ds Θ_{1}(K, d(x, u))
= (d − 1) ∫_{ℝ^{d}×S^{d−1}} ∫_{√\overline{r^{2}−a_{x}^{2}}}^{r} s^{d−2} ds Θ_{1}(K, d(x, u))
= ∫_{ℝ^{d}} (r^{d−1} − \sqrt{r^{2} − a_{x}^{2}}^{d−1}) C_{1}(K, dx).

Lemma 10 with n = d − 1 and a = a_{x} yields

I_{C}(r) ≤ \frac{d−1}{2} ∫_{ℝ^{d}} a_{x}^{2} C_{1}(K, dx)r^{d−3}

for d ≥ 3, and

I_{C}(r) ≤ \frac{1}{2} ∫_{ℝ^{d}} a_{x}^{2} C_{1}(K, dx)r^{−1} + \frac{\sqrt{2}}{4} ∫_{ℝ^{d}} a_{x}^{4} C_{1}(K, dx)r^{−3}

for d = 2. In view of (6) this shows the assertion.

We now derive the corresponding asymptotic lower bound for I_{C}(r)r^{−(d−3)}.

Lemma 13. For any compact C ⊆ ℝ^{d} we have

lim inf_{r→∞} \frac{I_{C}(r)}{r^{d−3}} ≥ \frac{ω_{d−1}}{2} V_{1}^{(3)}(C).  (23)

Proof. Let K be the convex hull of C. For x ∈ ℝ^{d} set τ_{x} = (q_{x}^{C} −p_{x}^{C})/∥q_{x}^{C} −p_{x}^{C}∥ if p_{x}^{C} ≠ q_{x}^{C}, and set τ_{x} = o, otherwise. The following arguments do not depend on the orientation of τ_{x}. For (x, u) ∈ N (K), ϵ ≥ 0 and δ > 0 we denote the indicator of the following event by ζ(x, u, ϵ, δ): for any y ∈ C the implication

⟨x − y, u⟩ ≤ ϵ ⇒ ⟨y, τ_{x}⟩ < ⟨p_{x}^{C}, τ_{x}⟩ + δ or ⟨y, τ_{x}⟩ > ⟨q_{x}^{C}, τ_{x}⟩ − δ

holds. Note that if p_{x}^{C} = q_{x}^{C} (and in particular if x ∈ C), we have ζ(x, u, ϵ, δ) = 1.

Fix (x, u) ∈ N (K) with x ∉ C, and numbers δ > 0, ϵ, s ≥ 0. Put

a_{x} = d(\{p_{x}^{C}, q_{x}^{C}\}, x) ≤ \frac{diam C}{2}

and r_{ϵ} = max{(diam C)^{2}/(4ϵ), (diam C)/2}. In order to find a lower bound for I_{C}(r), we will first show the inequality

ζ(x, u, ϵ, δ)1_{\{\sqrt{r^{2}−((a_{x}−δ)^{+})^{2}}≤s\}} ≤ 1_{\{d(C,x+su)≥r\}},  (24)

for all r ≥ r_{ϵ}. So assume ζ(x, u, ϵ, δ) = 1 and \sqrt{r^{2} − ((a_{x} − δ)^{+})^{2}} ≤ s. Let y ∈ C. If ⟨x − y, u⟩ ≤ ϵ, then ζ(x, u, ϵ, δ) = 1 implies

d(y, x + su)^{2} ≥ s^{2} + ⟨x − y, τ_{x}⟩^{2} ≥ s^{2} + ((a_{x} − δ)^{+})^{2} ≥ r^{2}.

If ⟨x − y, u⟩ > ϵ, then

d(y, x + su)^{2} ≥ (⟨x − y, u⟩ + s)^{2} ≥ (ϵ + \sqrt{r^{2} − ((a_{x} − δ)^{+})^{2}} )^{2} ≥ r^{2}

for all r ≥ r_{ϵ} by Lemma 10. So d(C, x + su) ≥ r for all r ≥ r_{ϵ}, which completes the proof of (24). Now let ϵ, δ > 0. For all r ≥ r_{ϵ} Lemma 8 and inequality (24) imply

I_{C}(r) = (d − 1) ∫_{(bd K)×A_{1}^{C}} ∫_{0}^{r} 1_{{d(C,x+su)≥r}}s^{d−2} ds Θ_{1}(K, d(x, u))

≥ (d − 1) ∫_{(bd K)×A_{1}^{C}} ∫_{√\overline{r^{2}−((a_{x}−δ)^{+})^{2}}}^{r} ζ(x, u, ϵ, δ)s^{d−2} ds Θ_{1}(K, d(x, u))

= ∫_{(bd K)×A_{1}^{C}} ζ(x, u, ϵ, δ)(r^{d−1} − \sqrt{r^{2} − ((a_{x} − δ)^{+})^{2}}^{d−1})Θ_{1}(K, d(x, u)).

Since

lim_{r→∞} \frac{r^{d−1} − \sqrt{r^{2} − ((a_{x} − δ)^{+})^{2}}^{d−1}}{r^{d−3}} = \frac{d−1}{2}((a_{x} − δ)^{+})^{2}

due to Lemma 10, Fatou’s lemma gives

lim inf_{r→∞} \frac{I_{C}(r)}{r^{d−3}} ≥ \frac{d−1}{2} ∫_{(bd K)×A_{1}^{C}} ζ(x, u, ϵ, δ) ((a_{x} − δ)^{+})^{2} Θ_{1}(K, d(x, u)).

Documenta Mathematica 18 (2013) 275–295

Now we first let ϵ → 0 using lim_{ϵ→0} ζ(x, u, ϵ, δ) = ζ(x, u, 0, δ), and the monotone convergence theorem. Since ζ(x, u, 0, δ)((a_{x} − δ)^{+})^{2} ≤ (diam C)^{2}, we can use the dominated convergence theorem to let δ → 0, and get

lim inf_{r→∞} \frac{I_{C}(r)}{r^{d−3}} ≥ \frac{d−1}{2} ∫_{(bd K)×A_{1}^{C}} lim_{δ→0} ζ(x, u, 0, δ)a_{x}^{2} Θ_{1}(K, d(x, u)).

For all (x, u) ∈ (bd K) × A_{1}^{C}, we have lim_{δ→0} ζ(x, u, 0, δ) = 1, and (23) follows using (6) and Lemma 8.

Theorem 2 now follows directly from Lemmas 11, 12, and 13.

3.3 Proof of Proposition 3

Without loss of generality, may assume that f(r) ≥ \frac{1}{r} holds for all r ∈ (0, ∞). Put g : [0, 1] → (−∞, 0], x ↦ −x/(6f^{−1}(x)), and let S = g(1) − 1,

C = {(x, g(x)) : x ∈ [0, 1]} ∪ {(x, g(2 − x)) : x ∈ [1, 2]} ∪ {(0, S), (2, S)}

and

C^{0} = {(0, 0), (2, 0), (0, S), (2, S)}.

For any r ∈ ℝ, large enough that f(r) ≤ 1/3, we have r ≥ \frac{1}{f(r)} ≥ 3 and

V_{2}(C_{r}) − V_{2}(C_{r}^{0}) ≥ \begin{aligned}∫_{0}^{1} ( max\{y ∈ ℝ & : (x, y) ∈ C_{r}\} \\ & − max\{y ∈ ℝ : (x, y) ∈ C_{r}^{0}\}) dx \end{aligned}

≥ \begin{aligned}∫_{0}^{1} ( max\{y ∈ ℝ & : ∥(x, y) − (f(r), g(f(r)))∥ ≤ r\} \\ & − max\{y ∈ ℝ : (x, y) ∈ C_{r}^{0}\}) dx \end{aligned}

= ∫_{0}^{1} (\sqrt{r^{2} − (x − f(r))^{2}} + g(f(r)) − \sqrt{r^{2} − x^{2}}) dx

≥ ∫_{0}^{1} (\frac{2xf(r) − f(r)^{2}}{2\sqrt{r^{2} − (x − f(r))^{2}}} + g(f(r))) dx

by using Lemma 10 with r, a, and n replaced by \sqrt{r^{2} − (x − f(r))^{2}}, \sqrt{2xf(r) − f(r)^{2}}, and 1, respectively. Since g(f(r)) = −\frac{f(r)}{6r}, this integral can be estimated from below by

∫_{0}^{1} \frac{2xf(r) − f(r)^{2}}{2r} − \frac{f(r)}{6r} dx = \frac{f(r)}{2r} − \frac{f(r)^{2}}{2r} − \frac{f(r)}{6r} ≥ \frac{f(r)}{6r}.

Documenta Mathematica 18 (2013) 275–295

Observing that K = conv C = conv C^{0} and V_{1}^{(3)}(C) = V_{1}^{(3)}(C^{0}) we conclude from Proposition 1 that there is a constant c_{1} ≥ 0 with

V_{1}^{(3)}(C)\frac{1}{r} − (V_{2}(K_{r}) − V_{2}(C_{r}))

= −(V_{2}((conv C^{0})_{r}) − V_{2}(C_{r}^{0}) − V_{1}^{(3)}(C^{0})\frac{1}{r}) + (V_{2}(C_{r}) − V_{2}(C_{r}^{0}))

≥ −\frac{c_{1}}{r^{3}} + \frac{f(r)}{6r}

≥ \frac{f(r)}{12r}

for all sufficiently large r. Hence

V_{1}^{(3)}(C) − \frac{V_{2}(K_{r}) − V_{2}(C_{r})}{r^{−1}} ≥ \frac{1}{12}f(r)

for all sufficiently large r and Proposition 3 is shown. Remark 14. In order to show a statement analogous to Proposition 3 in higher dimensions, one can consider bodies of revolution.

3.4 Proof of Theorem 4

Theorem 4 is a consequence of Lemmas 11, 12 and the following result.

Lemma 15. For any finite set C ⊆ ℝ^{d} there is a constant c = c(C) > 0 such that

I_{C}(r) ≥ \frac{ω_{d−1}}{2} V_{1}^{(3)}(C)r^{d−3} − cr^{d−4}  (25)

for all sufficiently large r.

Proof. We have

I_{C}(r) = (d − 1) ∫_{(bd K)×S^{d−1}} ∫_{0}^{r} 1_{(C_{r})^{C}}(x + su)s^{d−2} ds Θ_{1}(K, d(x, u)).

As

1_{(C_{r})^{C}}(x + su) = 1_{({p_{x}^{C},q_{x}^{C}}_{r})^{C}}(x + su)(1 − 1_{(C\{p_{x}^{C},q_{x}^{C}})_{r}}(x + su))

≥ 1_{({p_{x}^{C},q_{x}^{C}}_{r})^{C}}(x + su)(1 − ∑_{y∈C} 1_{[0,r]}(∥x + su − y∥)),

and

1_{(\{p_{x}^{C},q_{x}^{C}\}_{r})^{C}}(x + su) = 1_{(\sqrt{r^{2}−a_{x}^{2}},∞)}(s)

(with a_{x} = d({p_{x}^{C}, q_{x}^{C}}, x)), we get

I_{C}(r) ≥ ∫_{(bd K)×S^{d−1}} (r^{d−1} − \sqrt{r^{2} − a_{x}^{2}}^{d−1}) Θ_{1}(K, d(x, u)) − ∑_{y∈C} J_{C}(r, y). (26)

Here,

J_{C}(r, y) = (d − 1) ∫_{(bd K)×S^{d−1}} ∫_{0}^{r} 1_{({p_{x}^{C},q_{x}^{C}}_{r})^{C}}(x + su)×  (27)
× 1_{[0,r]}(∥x + su − y∥)s^{d−2} ds Θ_{1}(K, d(x, u)).

By Lemma 16 below, J_{C}(r, y) = O(r^{d−4}) for all y ∈ C, as r → ∞, and thus the second term on the right hand side of (26) is O(r^{d−4}). Lemma 10 with n = d − 1 and a = a_{x} together with (6) shows that the first term on the right hand side of (26) is bounded from below by

\frac{ω_{d−1}}{2} V_{1}^{(3)}(C)r^{d−3} − c_{d−1}κ_{d−1}V_{1}^{(5)}(C)r^{d−5}

for all sufficiently large r. Hence (25) follows from (26).

Lemma 16. Let C ⊆ ℝ^{d} be finite and y ∈ C. Then there is a constant c = c(C, y) > 0 such that

J_{C}(r, y) ≤ cr^{d−4}

for all sufficiently large r, where J_{C}(r, y) is defined by (27).

Proof. From (27) and (16) we get

J_{C}(r, y) = ∑_{e∈F_{1}^{∗}(K)} ∫_{e} ∫_{n(K,e)} ∫_{0}^{r} ζ(x + su)s^{d−2} ds dH^{d−2}(u) dH^{1}(x).

where ζ(z) = 1_{((relbd e)_{r})^{C}}(z)1_{[0,r]}(∥z − y∥). Note that the relative boundary relbd e of e consists just of the two endpoints of e. Spherical coordinates in hyperplanes orthogonal to e and Fubini’s theorem give

J_{C}(r, y) = ∑_{e∈F_{1}^{∗}(K)} ∫_{ℝ^{d}} 1_{e}(p(K, z))1_{n(K,e)}(u(K, z))ζ(z) dz.  (28)

Fix e ∈ F_{1}^{∗}(K) and let x_{1} and x_{2} be its endpoints. We assume without loss of generality that e contains the origin. Let g be the affine hull of e and let L be the affine hull of e and y. We may assume y ∉ g, since we cannot have y ∈ relint e and we have 1_{e}(p(K, z))1_{n(K,e)}(u(K, z))ζ(z) = 0 for all z ∈ ℝ^{d} if y ∈ g \ relint e. Let H^{+} be the closed half space containing e in its boundary with normal vector y −p(g, y), such that y ∉ H^{+}. Finally let V_{y} be the Voronoi cell of y with respect to the set {x_{1}, x_{2}, y}. The planar set

T = L ∩ V_{y} ∩ H^{+} ∩ (e + g^{⊥})

is either empty or a bounded triangle. For nonempty T let δ > 0 be the maximal distance from a point of T to {x_{1}, x_{2}}. If T = ∅ put δ = 0. If we can show that

1_{e}(p(K, z))1_{n(K,e)}(u(K, z))ζ(z) ≤ 1_{T} (z|L)1_{(\sqrt{r^{2}−δ^{2}},r]}(∥z|L^{⊥}∥) (29)

holds for all r > δ and z ∈ ℝ^{d}, then

∫ℝ^{d}1_{e}(p(K, z))1_{n(K,e)}(u(K, z))ζ(z) dz
≤ ∫_{L^{⊥}} ∫_{L} 1_{T} (y_{1})1_{(\sqrt{r^{2}−δ^{2}},r]}(∥y_{2}∥) dy_{1} dy_{2}
= V_{2}(T) · κ_{d−2}(r^{d−2} − \sqrt{r^{2} − δ^{2}}^{d−2})
≤ (d − 2)κ_{d−2}V_{2}(T)δ^{2}r^{d−4},

for all r > δ. The last inequality is evident in the case d = 2, and follows for d ≥ 3 from Lemma 10 with n = d − 2 and a = δ. Bounding all summands in (28) in such a way shows the assertion.

It remains to prove (29). Assume that the left hand side of (29) is one. Then p(K, z) ∈ e, u(K, z) ∈ n(K, e), ∥z−x_{1}∥ > r, ∥z−x_{2}∥ > r, and ∥z−y∥ ≤ r. The last three inequalities imply z ∈ V_{y} and z|L ∈ V_{y}|L = V_{y} ∩ L. The convexity of K and y ∈ K imply ⟨z − p(K, z), y − p(K, z)⟩ ≤ 0. Since both z − p(K, z) and y − p(g, y) are perpendicular to g, this gives z ∈ H^{+}. Finally,

z ∈ e + N(K, e) ⊆ e + g^{⊥}

gives z|L ∈ T. As ∥z|L^{⊥}∥ ≤ ∥z − y∥ ≤ r and

r^{2} < d({x_{1}, x_{2}}, z)^{2} = d({x_{1}, x_{2}}, z|L)^{2} + d(z|L, z)^{2} ≤ δ^{2} + ∥z|L^{⊥}∥^{2},

we have \sqrt{r^{2} − δ^{2}} < ∥z|L^{⊥}∥ ≤ r, and (29) is shown.

3.5 Proof of Theorem 5

We first show a key observation: If Condition (A) holds, then, for sufficiently large r, the part of the difference set K_{r} \ C_{r} that is projected on a face F is independent of the points of C outside F.

Lemma 17. Let C ⊆ ℝ^{d} be a finite set satisfying Condition (A). If K = conv C, m ∈ {0, …, d − 1}, and F ∈ F_{m}(K) then

(K_{r} \ C_{r}) ∩ (F + N(K, F)) = (F_{r} \ (C ∩ F)_{r}) ∩ (F + N(K, F))  (30)

for all sufficiently large r.

Proof. Since

K_{r} ∩ (F + N(K, F)) = F_{r} ∩ (F + N(K, F))

and (C ∩ F)_{r} ⊆ C_{r}, the set on the left-hand side is contained in the set on the right-hand side. To show the opposite inclusion, let V_{y} be the Voronoi cell of y ∈ C with respect to C, let S_{y} = V_{y} ∩ (F + N(K, F)) be the set of all the points in V_{y} with metric projection in F, and define

r_{0} = min{r ≥ 0 : S_{y} ⊆ (C ∩ F)_{r} for all y ∈ C with bounded S_{y}}.

Let r > r_{0} and assume

x ∈ (F_{r} \ (C ∩ F)_{r}) ∩ (F + N(K, F)).

Then, clearly, x ∈ K_{r}. Moreover, we have ∥x − y∥ > r for all y ∈ C ∩ F. As {V_{y} : y ∈ C} covers ℝ^{d}, there is a y ∈ C with x ∈ V_{y} ∩ (F + N(K, F)) = S_{y}. The definition of r_{0} and x ∉ (C ∩ F)_{r} imply that the closed convex set S_{y} is unbounded. Hence, there is a ray with direction v, say, completely contained in S_{y}. It follows that v ∈ N(K, F) and, as the ray is contained in V_{y}, that ⟨y, v⟩ ≥ h_{K}(v). Hence F and y are contained in a supporting hyperplane of K (with normal v), and thus they are contained in some facet G of K. As x′ = p(K, x) ∈ F, Condition (A) implies that there is a point y′ ∈ C ∩ F with ∥y − x′∥ ≥ ∥y′ − x′∥ and thus

d(C, x)^{2} = ∥y − x∥^{2} ≥ ∥y − x′∥^{2} + ∥x′ − x∥^{2}
≥ ∥y′ − x′∥^{2} + ∥x′ − x∥^{2} = ∥y′ − x∥^{2} > r^{2},

where the first equality is due to x ∈ V_{y}, the first inequality follows from ⟨y, x − x′⟩ ≤ ⟨x′, x − x′⟩ (see [8, Lemma 1.3.1]) and the last inequality is due to x ∉ (C ∩ F)_{r}. Hence x ∉ C_{r}, which completes the proof of (30).

We now prove Theorem 5. Let C ⊆ ℝ^{d} be a finite set that satisfies Condition (A), and set K = conv C. Assume r > diam C. Due to (14) with f = 1_{K_{r}\C_{r}}, (20), and (16) we have

V_{d}(K_{r}) − V_{d}(C_{r}) = ∑_{m=1}^{d−1} ∑_{F ∈F_{m}(K)} I_{F,m}  (31)

with

I_{F,m} = ∫_{F} ∫_{n(K,F)} ∫_{0}^{∞} 1_{K_{r}\C_{r}}(x + su)s^{d−m−1} ds dH^{d−m−1}(u) dH^{m}(x),

m ∈ {1, …, d − 1}, F ∈ F_{m}(K). For all sufficiently large r, Lemma 17 implies

I_{F,m} = ∫_{F} ∫_{n(K,F)} ∫_{0}^{∞} 1_{F_{r}\(C∩F)_{r}}(x + su)s^{d−m−1} ds dH^{d−m−1}(u) dH^{m}(x)

= ∫_{F} ∫_{n(K,F)} ∫_{√\overline{r^{2}−d(C∩F,x)^{2}}}^{r} s^{d−m−1} ds dH^{d−m−1}(u) dH^{m}(x)

= \frac{ω_{d−m}}{d − m}γ(F, K) ∫_{F} (r^{d−m} − \sqrt{r^{2} − d(C ∩ F, x)^{2}}^{d−m}) dH^{m}(x).

Put a_{F,x} = d(C ∩ F, x). The binomial series

r^{d−m} − \sqrt{r^{2} − a_{F,x}^{2}}^{d−m} = r^{d−m} ∑_{k=1}^{∞}(−1)^{k+1}\binom{(d − m)/2}{k}(\frac{a_{F,x}}{r} )^{2k}

Documenta Mathematica 18 (2013) 275–295

converges absolutely as r > diam C ≥ a_{F,x}. Hence

I_{F,m} = ∑_{k=1}^{∞}(−1)^{k+1}\binom{(d − m)/2}{k}κ_{d−m}γ(F, K) ∫_{F} a_{F,x}^{2k} dH^{m}(x)r^{d−m−2k}

= \begin{aligned}∑_{{n=2−(d−m), \atop 2|(n+d−m)}}^{∞} & (−1)^{(n+d−m+2)/2}\binom{(d − m)/2}{(n + d − m)/2}× \\ & × κ_{d−m}γ(F, K) ∫_{F} a_{F,x}^{n+d−m} dH^{m}(x) r^{−n}. \end{aligned}

Substitution into (31) and definition (4) gives

V_{d}(K_{r}) − V_{d}(C_{r}) = ∑_{m=1}^{d−1} ∑_{{n=2−(d−m), \atop 2|(n+d−m)}}^{∞}(−1)^{(n+d−m+2)/2}\binom{(d − m)/2}{(n + d − m)/2}×

× κ_{d−m}V_{m}^{(n+d)}(C) r^{−n}

= ∑_{n=3−d}^{∞} ∑_{{m=1, \atop 2|(n+d−m)}}^{min\{d−1,n+d−2\}}(−1)^{(n+d−m+2)/2}\binom{(d − m)/2}{(n + d − m)/2}×

× κ_{d−m}V_{m}^{(n+d)}(C) r^{−n}.

If n is positive and even, the inner sum only involves integers m with the same parity as d, and (d − m)/2 < (n + d − m)/2 are both integers implying \binom{(d−m)/2}{(n+d−m)/2} = 0. Hence all coefficients of r^{−n} vanish for even n > 0. This concludes the proof of Theorem 5.

3.6 The example of the unit cube.

We show (11). In arbitrary dimension d the number of i-dimensional faces of [0, 1]^{d} is

#F_{i}([0, 1]^{d}) = 2^{d−i}\binom{d}{d − i}.

Using orthogonal projections and symmetry we get

γ(F, [0, 1]^{d}) = γ({o}, [0, 1]^{d−i}) = (#F_{0}([0, 1]^{d−i}))^{−1} = 2^{−(d−i)}

for any F ∈ F_{i}([0, 1]^{d}), i = 0, …, d − 1. Thus definition (4) and a symmetry argument give

V_{i}^{(m+i)}(C^{d}) = \binom{d}{d − i} ∫_{[0,1]^{i}} d(C^{i}, x)^{m}dx

= 2^{i}\binom{d}{d − i} ∫_{[0,1/2]^{i}} ∥x∥^{m}dx

= 2^{−m}\binom{d}{d − i} ∫_{[0,1]^{i}} ∥x∥^{m}dx.

This implies

V_{1}^{(m+1)}(C^{d}) = \frac{d}{(m + 1)2^{m}}  (32)

for m ≥ 0, and, introducing polar coordinates,

V_{2}^{(m+2)}(C^{d}) = 2^{−m}\binom{d}{d − 2}2 ∫_{0}^{π/4} ∫_{0}^{1/ cos φ} ∥r · (cos φ, sin φ)∥^{m}r dr dφ

= \frac{d(d − 1)}{(m + 2)2^{m}} ∫_{0}^{π/4} cos^{−(m+2)}(φ) dφ.

We put d_{n} := ∫_{0}^{π/4} cos^{−n}(φ) dφ. Integrating ∫_{0}^{π/4} cos^{−(n+1)}(φ) cos(φ) dφ by parts, we obtain the recurrence relation

(n + 1)d_{n+2} = 2^{n/2} + nd_{n}, n ≥ 0,

with starting value d_{0} = \frac{π}{4} . Induction gives

d_{2m} = 2^{m−1} \frac{(m − 1)!}{(2m − 1)!!} ∑_{i=0}^{m−1} \frac{(2i − 1)!!}{i!},

and we arrive at

V_{2}^{(2m+2)}(C^{3}) = \frac{3}{(m + 1)4^{m}} d_{2m+2}
= \frac{3(m!)}{(2m + 1)!! (m + 1)2^{m}} ∑_{i=0}^{m} \frac{(2i − 1)!!}{i!}.  (33)

As C^{3} satisfies Condition (A), (11) follows by substituting (32) with d = 3, m = 2, and (33) into (10).

References

[1] H. Federer (1959): Curvature measures. Trans. Amer. Math. Soc. 93, 418-491.

[2] H. Federer (1969): Geometric Measure Theory. Springer, Heidelberg.

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[5] J. Kampf (2012): The parallel volume at large distances. Geometriae Dedicata 160, 47–70.

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Jürgen Kampf Department of Mathematics TU Kaiserslautern Postbox 3049 D-67653 Kaiserslautern Germany Tel. 0049 631 205 4825 kampf@mathematik.unikl.de Markus Kiderlen Department of Mathematical Sciences University of Aarhus Ny Munkegade 118 DK-8000 Aarhus C Denmark Tel. 0045 87 16 01 05 kiderlen@imf.au.dk 296

Documenta Mathematica 18 (2013)