Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 43 (2002), No. 1, 135-152.

On the Affine Convexity of Convex Curves and Hypersurfaces

Kurt Leichtweiß

Mathematisches Institut B, Universität Stuttgart Pfaffenwaldring 57, D-70550 Stuttgart

1. Introduction

It is well known that there are interesting analogies between the euclidean geometry and the equiaffine geometry of plane curves. As an example we would like to mention that there exists an affine rectification for arbitrary convex curves analogous to the euclidean rectification by inscribing parabola polygons into the (generalized) tangent bundle of the curve (see [7]). Hereby each parabola arc (with vanishing affine curvature k) connecting its final support elements replaces a segment (with vanishing euclidean curvature κ) connecting its endpoints.

Now a closed straight polygon Π in the euclidean plane is a convex curve if and only if the lines carrying the segments are supporting Π. In analogy to this we shall call a closed convex parabola polygon 𝒫 affinely convex if and only if the parabolas carrying the segments of 𝒫 are supporting 𝒫. After a characterization of such affinely convex parabola polygons in Section 2 we consider in Section 3 as limiting case smooth ovals C in the plane and define them to be affinely convex if and only if every hyperosculating parabola of C is supporting C. One main result is now the characterization of affinely convex ovals by the condition k ≧ 0 in analogy to the euclidean characterization of smooth convex curves as simply closed curves with κ ≧ 0. (We owe partial results in this direction to T. Carleman [3]). Moreover, each parabola arc connecting two support elements of an affinely convex oval either is a part of this oval or it does not intersect the latter in its interior, in analogy to the euclidean case. This fact is an improvement of an old result of K. Reidemeister [11].

Unfortunately ovaloids F in the d-space (d ≧ 3) don’t possess hyperosculating paraboloids in general. Therefore the notion of affine convexity may be defined for them in Section 4 only in an analytic way, namely by the claim that their affine curvatures k_{1}, …, k_{d−1} are nonnegative everywhere. But it turns out that this definition (made in analogy to the euclidean geometry) has interesting consequences: The affine surface area of affinely convex ovaloids is a strictly monotone increasing (with respect to inclusion) and a continuous (with respect

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 2002 Heldermann Verlag

to the Hausdorff topology) functional, properties which are not valid for general ovaloids. But these properties are also valid for affinely convex ovals and parabola polygons. All these things follow from the fact that the convex domains resp. bodies bounded by these curves resp. hypersurfaces belong to a special class for which C. Petty [10] had proved the indicated properties.

2. Affinely convex parabola polygons

We begin with

Definition 2.1. The union of a finite set of parabola arcs 𝒫_{l−1l} in the affine plane ℝ_{2}, situated on the pairwise different parabolas P_{l−1l} and determined by their final support elements (points and tangent lines) (x_{l−1}, t_{l−1}) and (x_{l}, t_{l}) (l = 1, …, k) with

x_{k} = x_{0}, t_{k} = t_{0},  (1)

is called a parabola polygon 𝒫:

𝒫 := ⋃_{l=1}^{k} 𝒫_{l−1l}  (2)

if 𝒫 is a simply closed convex curve.

We say that 𝒫 is inscribed into a simply closed plane convex curve C if all support elements (x_{l}, t_{l}) of 𝒫 belong to the generalized tangent bundle 𝒯(C) of C consisting of all support elements of C (points and support lines) (l = 1, …, k).

Definition 2.2. A parabola polygon 𝒫 is affinely convex if any parabola P_{l−1l} of 𝒫 supports the curve 𝒫, i.e. if 𝒫 is contained in the closed convex halfspace bounded by P_{l−1l} (l = 1, …, k).

For example a regular parabola polygon, inscribed into a circle, is affinely convex as well as any of its equiaffine images (see Fig. 1 for a regular parabola triangle). Otherwise, if two adjacent arcs of a parabola polygon 𝒫 are inscribed into one branch of a hyperbola H then 𝒫 cannot be affinely convex (see Fig. 2).

Figure 1 Figure 2 Now it is possible to characterize affinely convex parabola polygons in the following manner:

Proposition 2.3. A parabola polygon 𝒫 is affinely convex if and only if the following conditions are fulfilled:

i) The parabolas P_{l−1l} and P_{ll+1} of the arcs 𝒫_{l−1l} and 𝒫_{ll+1} of 𝒫 do not only touch at x_{l} but also osculate at this point of second order, i.e. they have three infinitesimally neighbouring points in common there (l = 1, …, k). ii) The direction angles φ_{l−1l} of the axes of the parabolas P_{l−1l} of 𝒫 are ordered by (w.l.o.g.)

0 ≦ φ_{01} < φ_{12} < ⋯ < φ_{k−1k} < 2π  (3)

if the arcs 𝒫_{l−1l} of 𝒫 are ordered in the positive sense (l = 1, …, k).

Proof. At first we shall show that the conditions i) and ii) are necessary. Indeed, as by the affine convexity of 𝒫 the parabola P_{l−1l} is supporting in particular the arc 𝒫_{ll+1} of 𝒫 the euclidean curvature κ^{(l−1l)}(x_{l}) of P_{l−1l} at x_{l} cannot exceed the euclidean curvature κ^{(ll+1)}(x_{l}) of 𝒫_{ll+1} at x_{l}:

κ^{(l−1l)}(x_{l}) ≦ κ^{(ll+1)}(x_{l}).  (4)

In the same way, as P_{ll+1} is supporting in particular 𝒫_{l−1l}, we find

κ^{(ll+1)}(x_{l}) ≦ κ^{(l−1l)}(x_{l})  (5)

such that by (4) and (5)

κ^{(l−1l)}(x_{l}) = κ^{(ll+1)}(x_{l})

or – equivalently –

\frac{d^{2}x^{(l−1l)}}{dσ^{2}} (σ_{l}) = \frac{d^{2}x^{(ll+1)}}{dσ^{2}} (σ_{l})  (6)

besides

x^{(l−1l)}(σ_{l}) = x^{(ll+1)}(σ_{l}), \frac{dx^{(l−1l)}}{dσ} (σ_{l}) = \frac{dx^{(ll+1)}}{dσ} (σ_{l})

which yields property i) (x = x^{(l−1l)}(σ) resp. x = x^{(ll+1)}(σ) are suitable representations of P_{l−1l} resp. P_{ll+1} with the help of the euclidean arclength σ).

Now it is clear that the (different) conics in the projective plane ℙ_{2} := ℝ_{2}∪{l_{∞}}, resulting after completion of the parabolas P_{l−1l} and P_{ll+1} in ℝ_{2} by their improper points i_{l−1l} and i_{ll+1} (where they touch the improper line l_{∞}), may have only one further point in common besides the triple point x_{l} (l = 1, …, k), not lying on l_{∞}. This additional point of intersection must exist by topological reasons such that we have the situation as indicated in Fig. 3 where the axes a_{l−1l} resp. a_{ll+1} of P_{l−1l} resp. P_{ll+1} passing through the point x_{l} with direction angles φ_{l−1l} resp. φ_{ll+1} satisfy

φ_{l−1l} < φ_{ll+1} (l = 1, …, k).

A suitable numeration of the parabola arcs then shows the validity of ii).

Figure 3

Conversely, we will assume now that the parabola polygon 𝒫 fulfils the conditions i) and ii), and we want to prove that 𝒫 must be affinely convex. For this reason we fix the parabola P_{l−1l} carrying the arc 𝒫_{l−1l} of 𝒫 and consider at first the part P_{l−1l}^{+} of P_{l−1l} connecting the points x_{l} and i_{l−1l} of P_{l−1l} in the positive sense. Then P_{l−1l}^{+} meets the segment 𝒫_{ll+1} ∪ P_{ll+1}^{+} of P_{ll+1} between x_{l} and i_{ll+1} only at x_{l} since the conics P_{l−1l} and P_{ll+1} have except the triple point x_{l} (see i)) only one further point in common which lies out of P_{l−1l}^{+} because of ii) (compare Fig. 3!).

Figure 4

Now we consider the point v_{l}^{+} of 𝒫 with (oriented) tangent line parallel to the axis a_{l−1l} of P_{l−1l} through x_{l}. If v_{l}^{+} lies on 𝒫_{ll+1} we are sure that the arc 𝒫_{l}^{+} of 𝒫 from x_{l} to v_{l}^{+} (in a positive sense) does not meet the part P_{l−1l}^{−} of P_{l−1l} from x_{l} to i_{l−1l} (in a negative sense) with exception of x_{l} because 𝒫_{l}^{+} and P_{l−1l}^{−} are separated by a_{l−1l}. Therefore P_{l−1l} supports the part 𝒫_{l}^{+} of 𝒫. In the other case where v_{l}^{+} lies on 𝒫_{mm+1} with l < m we iterate the preceding considerations from l to m and see that P_{l−1l} supports 𝒫_{l}^{+} too. In the same way we find that that P_{l−1l} supports the arc 𝒫_{l}^{−} of 𝒫 from x_{l} to the point v_{l}^{−} (in a negative sense) with an oriented line in the opposite direction to the direction of a_{l−1l}. Since it is trivial that the remainder 𝒫 \ (𝒫_{l}^{+} ∪ 𝒫_{l}^{−}) of 𝒫 is also supported by P_{l−1l} we finally know that P_{l−1l} supports 𝒫 (see Fig. 4). As this is true for l = 1, …, k the parabola polygon 𝒫 must be affinely convex by Definition 2.2 which completes the proof of Proposition 2.3. □

3. Affinely convex ovals

In this section we consider ovals instead of parabola polygons, so to speak as limiting case of the latter ones. An oval C is a simply closed plane curve x = x(σ) of class 𝒞_{2} in ℝ_{2} with positive curvature

κ := (\frac{dx}{dσ}, \frac{d^{2}x}{dσ^{2}})  (7)

everywhere (σ euclidean arclength parameter of C). It is well-known that it is possible to introduce for C an equiaffine arclength parameter s by

s := ∫_{σ_{0}}^{σ} κ^{\frac{1}{3}} dσ  (8)

whence by (7) for a curve of class 𝒞_{3}

(\frac{dx}{ds}, \frac{d^{2}x}{ds^{2}}) = 1.  (9)

Supposing now that C be of class 𝒞_{4} with respect to σ the curve C is of class 𝒞_{3} with respect to s and differentiation of (9) yields

(\frac{dx}{ds}, \frac{d^{3}x}{ds^{3}}) = 0.  (10)

Thus, after introduction of the so-called affine normal vector

y := \frac{d^{2}x}{ds^{2}}  (11)

we may define the coefficient k in the relation

\frac{d^{3}x}{ds^{3}} = \frac{dy}{ds} = −k \frac{dx}{ds}  (12)

(see (10)) as affine curvature k of the oval C (in analogy to the equation \frac{dn}{dσ} = −κ \frac{dx}{dσ} for the euclidean curvature κ of C where n is the inner unit normal vector).

As limit case we have now to replace the parabola P_{l−1l} containing the parabola arc from the support element (x(s_{l−1}), t(s_{l−1})) of C to the support element (x(s_{l}), t(s_{l})) with the join a_{l} of t(s_{l−1}) ∩ t(s_{l}) and \frac{1}{2}(x(s_{l−1}) + x(s_{l})) as an axis by the parabola P_{s_{0}} hyperosculating C at x(s_{0}) of third order with the axis a_{s_{0}} through x(s_{0}). This is obvious because P_{s_{0}} has four infinitesimally neighbouring points in common with C there. P_{s_{0}} may be represented by

x = z(s) := x(s_{0}) + (s − s_{0})\frac{dx}{ds}(s_{0}) + \frac{1}{2}(s − s_{0})^{2} \frac{d^{2}x}{ds^{2}}(s_{0})  (13)

where s is also an affine arclength parameter because of

(\frac{dz}{ds}, \frac{d^{2}z}{ds^{2}} ) = (\frac{dx}{ds}(s_{0}) + (s − s_{0})\frac{d^{2}x}{ds^{2}}(s_{0}), \frac{d^{2}x}{ds^{2}}(s_{0})) = (\frac{dx}{ds}(s_{0}), \frac{d^{2}x}{ds^{2}}(s_{0})) = 1  (14)

(see (9)). It has indeed the property of hyperosculation since

z(s_{0}) = x(s_{0}), \frac{d^{ν}z}{ds^{ν}} (s_{0}) = \frac{d^{ν}x}{ds^{ν}}(s_{0}) (ν = 1, 2)  (15)

whence by (8),

\frac{dx}{ds} = κ^{− \frac{1}{3}} · \frac{dx}{dσ}, \frac{d^{2}x}{ds^{2}} = −\frac{1}{3}κ^{− \frac{5}{3}} \frac{dκ}{dσ} · \frac{dx}{dσ} + κ^{\frac{1}{3}} · n

and

\frac{d^{2}x}{dσ^{2}} = κn, \frac{d^{3}x}{dσ^{3}} = −κ^{2} · \frac{dx}{dσ} + \frac{dκ}{dσ} · n

(n inner unit normal vector) we get

z(s_{0}) = x(s_{0}), \frac{d^{μ}z}{dσ^{μ}} (s_{0}) = \frac{d^{μ}x}{dσ^{μ}}(s_{0}) (μ = 1, 2, 3)  (16)

(compare condition (6) for the osculation of P_{l−1l} and P_{ll+1} !). These facts motivate in analogy to Definition 2.2 the

Definition 3.1. An oval C of class 𝒞_{4} is called affinely convex if any hyperosculating parabola P_{s_{0}} of C supports C (in the same sense as in Definition 2.2).

We have to mention that 1940 T. Carleman [3] made the same definition (in a local sense) for ovals of class 𝒞_{4} which were called courbes paraboliquement convexes. Using the representation ξ_{2} = ξ_{2}(ξ_{1}) for the points x = (ξ_{1}, ξ_{2}) of C this author found as a necessary condition for such a curve

\frac{d^{2}}{dξ_{1}^{2}} ((\frac{d^{2}ξ_{2}}{dξ_{1}^{2}} )^{− \frac{2}{3}} ) ≦ 0.  (17)

Since we have

−\frac{1}{2} \frac{d^{2}}{dξ_{1}^{2}} ((\frac{d^{2}ξ_{2}}{dξ_{1}^{2}} )^{− \frac{2}{3}} ) = k  (18)

(see [1], p. 14 (83)) this condition (17) is equivalent to k ≧ 0 for all points of C. Carleman even proved in addition that conversely the stronger condition k > 0 for the oval C implies that all its hyperosculationg parabolas P_{s_{0}} have the property

C ∩ P_{s_{0}} = {x(s_{0})}.  (19)

We shall now prove the stronger

Theorem 3.2. An oval C of class 𝒞_{4} is affinely convex if and only if the affine curvature k of C is nonnegative everywhere:

k ≧ 0.  (20)

Proof. In the first part we show that the condition (20) is necessary. For this purpose we use the representation

(z − x(s_{0}), \frac{d^{2}x}{ds^{2}}(s_{0}))^{2} + 2(z − x(s_{0}), \frac{dx}{ds}(s_{0})) ≦ 0  (21)

of the closed convex region bounded by P_{s_{0}} following from (13) and (9). As C is supported by P_{s_{0}} as an affinely convex oval we have

A(s) := (x(s) − x(s_{0}), \frac{d^{2}x}{ds^{2}}(s_{0}))^{2} + 2(x(s) − x(s_{0}), \frac{dx}{ds}(s_{0})) ≦ 0  (22)

for all s whence in particular using (12)

A(s_{0}) = \frac{dA}{ds}(s_{0}) = \frac{d^{2}A}{ds^{2}}(s_{0}) = \frac{d^{3}A}{ds^{3}}(s_{0}) = 0, \frac{d^{4}A}{ds^{4}}(s_{0}) = −6k(s_{0}) ^{1}.  (23)

We make now the assumption

\frac{d^{4}A}{ds^{4}}(s_{0}) > 0  (24)

such that because of \frac{d^{3}A}{ds^{3}}(s_{0}) = 0 the relation

\frac{d^{3}A}{ds^{3}}(s) > 0 (s_{0} < s < s_{0} + δ)  (25)

holds for a suitable δ > 0. On the other hand iterated application of the mean value theorem together with (22) and (23) yields

\frac{dA}{ds}(s_{0} + δ_{1}) ≦ 0, \frac{d^{2}A}{ds^{2}}(s_{0} + δ_{2}) ≦ 0 and \frac{d^{3}A}{ds^{3}}(s_{0} + δ_{3}) ≦ 0

1In order to get \frac{d^{4}A}{ds^{4}}(s_{0}) we have used the rule \frac{d}{ds}(f · g)(s_{0}) = f(s_{0}) · \frac{dg}{ds} (s_{0}) if f is continuous at s_{0} and g is differentiable at s_{0} with g(s_{0}) = 0.

for suitable 0 < δ_{3} < δ_{2} < δ_{1} < δ contradicting (25). Thus the assumption (24) is wrong with the consequence that by (23) indeed k(s_{0}) ≧ 0 is valid for every s_{0} ^{2}.

Now we prove that (20) is also sufficient for the oval C to be affinely convex. Therefore we introduce at first the affine evolvents E_{τ} of C with the representation

w(s, τ) := x(s) + (τ − s)\frac{dx}{ds}(s) + \frac{1}{2}(τ − s)^{2} \frac{d^{2}x}{ds^{2}}(s) (s_{0} ≦ s ≦ τ)  (26)

depending on a parameter τ such that τ − s_{0} is the affine arclength of the curve which is the union of the arc of C from x(s_{0}) to x(s) and the arc of P_{s} from x(s) to w(s, τ). E_{τ} is a curve of class 𝒞_{1} with respect to s which joins the points w(s_{0}, τ) and w(τ, τ) = x(τ). Moreover this curve penetrates each parabola P_{s} either in a stationary or in a transversal manner, a consequence of the relation

(\frac{∂w}{∂τ}(s, τ), \frac{∂w}{∂s}(s, τ))=(\frac{dx}{ds}(s)+(τ −s)\frac{d^{2}x}{ds^{2}}(s), \frac{1}{2}(τ −s)^{2} \frac{d^{3}x}{ds^{3}}(s))= \frac{1}{2}(τ −s)^{3}k(s) ≧ 0 (27)

Figure 5

(see (26) and (20)). Therefore we are sure that the arc C_{s_{0}}^{+} of C from x(s_{0}) to the point v^{+}(s_{0}) with (oriented) tangent line parallel to the axis a_{s_{0}} of P_{s_{0}} is supported by the part P_{s_{0}}^{+} of P_{s_{0}} from x(s_{0}) to its improper point i_{s_{0}} on the “convex side”. Moreover C_{s_{0}}^{+} does not meet the part P_{s_{0}}^{−} of P_{s_{0}} from i_{s_{0}} to x(s_{0}) with exception of x(s_{0}) because C_{s_{0}}^{+} and P_{s_{0}}^{−} are separated by a_{s_{0}}. Thus P_{s_{0}} supports the part C_{s_{0}}^{+} of C. In the same way we find that P_{s_{0}} also supports the arc C_{s_{0}}^{−} of C from x(s_{0}) to the point v^{−}(s_{0}) with (oriented) tangent line in the opposite direction to that of a_{s_{0}}. Finally P_{s_{0}} supports trivially the arc C \ (C_{s_{0}}^{+} ∪ C_{s_{0}}^{−}) of C and these facts altogether complete the proof of Theorem 3.2 if x(s_{0}) is arbitrarily chosen on C (see Fig. 5 and compare it with Fig. 4). □

2This can also be seen after application of a formula of J. Merza ([8], théorème 2): k(s_{0}) = lim_{s→s_{0}} \frac{8d̄(s)}{(s−s_{0})^{4}} where the (nonnegative) affine distance d̄(s) of x(s) from P_{s_{0}} is given by x(s) = z(s_{1}) + d̄(s)y(s_{0}) if the curve C : x = x(s) is sufficiently smooth.

Corollary 3.3. An oval C of class 𝒞_{4} is affinely convex if and only if the direction angles φ(s) of the axes a_{s} of the hyperosculating parabolas P_{s} of C have the property

\frac{dφ}{ds}(s) ≧ 0  (28)

(compare Proposition 2.3 ii)).

This is an immediate consequence of the known fact that the direction of the axes of a parabola P_{s} equals the direction of the (constant) affine normal vectors of P_{s} which by (13) and (15) equals the direction of the affine normal vector y(s) of C at x(s). This remark namely yields

\frac{dφ}{ds} = \frac{d}{ds} (arctan \frac{η_{2}}{η_{1}} ) = \frac{(y, \frac{dy}{ds})}{η_{1}^{2} + η_{2}^{2}} = \frac{k}{η_{1}^{2} + η_{2}^{2}} ≧ 0

for all s (see (12)) for y = (η_{1}, η_{2}) ≠ 0.

For example every ellipse (with affine normals intersecting in its midpoint lying on its convex side) is affinely convex, and every arc of a branch of a hyperbola (with affine normals intersecting in its midpoint lying on its concave side) cannot be part of an affinely convex oval. Now affinely convex ovals have the following “convexity property” which was an important tool in the early considerations of P. Böhmer [2] and H. Mohrmann [9] for (closed and open) ovals with k > 0 or k < 0 everywhere (see also W. Blaschke [1], p. 47–49):

Theorem 3.4. If 𝒫_{s_{0}s_{1}} is the parabola arc with the support elements (x(s_{0}), t(s_{0})) and (x(s_{1}), t(s_{1})) of an affinely convex oval C of class 𝒞_{4} as final support elements (s_{0} < s_{1}, ∡(t(s_{0}), t(s_{1})) < π) then either

i) 𝒫_{s_{0}s_{1}} ∩ C = {x(s_{0}), x(s_{1})} or ii) 𝒫_{s_{0}s_{1}} is a part of C.

Proof. For the proof of Theorem 3.4 we carry out a slight modification together with an improvement of a proof of K. Reidemeister [11]. For this purpose we introduce in the plane ℝ_{2} an affine coordinate system {ξ_{1}, ξ_{2}} in such a manner that we get x(s_{0}) = (0, 1), x(s_{1}) = (1, 0) and t(s_{0}) : ξ_{1} = 0, t(s_{1}) : ξ_{2} = 0. Then 𝒫_{s_{0}s_{1}} may be represented by

ξ_{1} = ζ_{1}(τ) := τ^{2}, ξ_{2} = ζ_{2}(τ) := (1 − τ)^{2} (0 ≦ τ ≦ 1),  (29)

and its affine arclength parameter equals 4^{\frac{1}{3}} τ. Furthermore the arc 𝒞 of C between x(s_{0}) and x(s_{1}) may be represented by x = x(s) with s_{0} ≦ s_{1} or

ξ_{1} = ξ_{1}(τ), ξ_{2} = ξ_{2}(τ) (0 ≦ τ ≦ 1)  (30)

with

τ := \frac{s − s_{0}}{L(𝒞)} (s_{0} ≦ s ≦ s_{1}, L(𝒞) := s_{1} − s_{0}).  (31)

Here ξ_{1} and ξ_{2} are functions of τ of class 𝒞_{3} on [0, 1] having the properties

ξ_{1}(0) = 0, ξ_{2}(0) = 1, ξ_{1}(1) = 1, ξ_{2}(1) = 0  (32)

and

\frac{dξ_{1}}{dτ} (0) = 0, \frac{dξ_{2}}{dτ} (1) = 0.  (33)

After introduction of f_{1} := ξ_{1} − ζ_{1}, f_{2} := ξ_{2} − ζ_{2}, two functions of class 𝒞_{3} on [0, 1], we get using (29), (32) and (33)

f_{1}(0) = f_{1}(1) = \frac{df_{1}}{dτ} (0) = 0,  (34)

f_{2}(0) = f_{2}(1) = \frac{df_{2}}{dτ} (1) = 0  (35)

as well as by (12), (20) and (31)

\frac{d^{3}f_{1}}{dτ^{3}} (τ) = \frac{d^{3}ξ_{1}}{dτ^{3}} (τ) = −k(τ)L(𝒞)^{2} \frac{dξ_{1}}{dτ} (τ) ≦ 0,  (36)

\frac{d^{3}f_{2}}{dτ^{3}} (τ) = \frac{d^{3}ξ_{2}}{dτ^{3}} (τ) = −k(τ)L(𝒞)^{2} \frac{dξ_{2}}{dτ} (τ) ≧ 0 (0 ≦ τ ≦ 1).  (37)

But (36) means that \frac{d^{2}f_{1}}{dτ^{2}} is a monotone decreasing function, either positive on [0, 1] or positive on [0, α), zero on [α, β] and negative on (β, 1] (0 ≦ α ≦ β ≦ 1) or negative on [0, 1]. Therefore the function f_{1} itself is either strictly convex or strictly convex on [0, α), linear on [α, β] and strictly concave on (β, 1] or strictly concave. But such a function only fits into the boundary conditions (34) if it is either positive in (0, 1) such that

ξ_{1}(τ) > τ^{2} (0 < τ < 1)  (38)

or if it vanishes identically such that

ξ_{1}(τ) ≡ τ^{2}  (39)

(see (29)). In the same way we find that the function f̄_{2}(τ) := f_{2}(1 − τ) (0 ≦ τ ≦ 1) which fulfils the same inequality and boundary conditions as f_{1} because of (37) and (35) either has the property f̄_{2}(τ) > 0 (0 < τ < 1) or f̄_{2}(τ) ≡ 0 such that by (29) either

ξ_{2}(τ) > (1 − τ)^{2} (0 < τ < 1) or  (40)

ξ_{2}(τ) ≡ (1 − τ)^{2}  (41)

holds.

Geometrically the results (38),(39) and (40),(41) may be interpreted as follows: In the case ξ_{1}(τ) > τ^{2} and ξ_{2}(τ) > (1 − τ)^{2} (0 < τ < 1) every point (ξ_{1}(τ′), ξ_{2}(τ′)) (τ′ = \frac{s′−s_{0}}{s_{1}−s_{0}} with s_{0} < s′ < s_{1}, see (31)) of 𝒞 lies in the open quadrant

Q_{τ′} := {(ξ_{1}, ξ_{2}) ∈ ℝ_{2}| ξ_{1} > τ′^{2}, ξ_{2} > (1 − τ′)^{2}}  (42)

of ℝ_{2} for which we have 𝒫_{s_{0}s_{1}} ∩ Q_{τ′} = ∅ whence

𝒫_{s_{0}s_{1}} ∩ 𝒞 = {x(s_{0}), x(s_{1})}  (43)

follows. As trivially 𝒫_{s_{0}s_{1}} ∩ (C \ 𝒞) = ∅ indeed i) holds. The same fact is true in the cases ξ_{1}(τ) > τ^{2} and ξ_{2}(τ) ≡ (1 − τ)^{2} as well as ξ_{1}(τ) ≡ τ^{2} and ξ_{2}(τ) > (1 − τ)^{2} for 0 < τ < 1. Since in the last case ξ_{1}(τ) ≡ τ^{2} and ξ_{2}(τ) ≡ (1 − τ)^{2} the parabola arc 𝒫_{s_{0}s_{1}} is a part of C such that ii) holds the proof of Theorem 3.4 is complete. □

Remark 3.5. We have proved more than claimed in i) or ii) of Theorem 3.4, namely: The arc 𝒞 of C between x(s_{0}) and x(s_{1}) lies in the closed (convex) region bounded by the segment x(s_{0})x(s_{1}) and the parabola arc 𝒫_{s_{0}s_{1}}. Moreover it should be mentioned that in this theorem 𝒞 may be replaced by an affinely convex oval arc from x(s_{0}) to x(s_{1}) with tangent lines at the endpoints intersecting within the closed triangle with the vertices x(s_{0}), x(s_{1}) and t(s_{0}) ∩ t(s_{1}) (i.e. (33) may be replaced by the inequalities \frac{dξ_{1}}{dτ} (0) ≧ 0, \frac{dξ_{2}}{dτ} (1) ≦ 0) (compare [1], p. 47–48).

4. Affinely convex ovaloids

Unfortunately it is not possible to extend Definition 3.1 of affinely convex ovals to ovaloids F (compact orientable hypersurfaces in ℝ_{d} (d > 2) of class 𝒞_{2} with positive Gauss curvature H_{d−1} everywhere and bijective Gauss map F → S^{d−1} because a hyperosculating paraboloid for F at a point x of F only exists in the case where the so-called cubic form A(x) of F at x vanishes (see [1], p. 107 (29) and p. 114 (78) for d = 3). Moreover the (elliptic) paraboloids are not the only equiaffine analogues for the hyperplanes in the euclidean differential geometry, the same role are playing more generally all the improper affine hyperspheres (see [1], p. 209– 210 for d = 3). For these reasons we shall define affinely convex ovaloids in a purely analytical manner keeping the consistency with the case d = 2 (see Theorem 3.2) by generalization of (20) to the case d > 2. Before doing so we note some basic facts of the equiaffine differential geometry of ovaloids F : x = x(u^{1}, …, u^{d−1}) in ℝ_{d} of class 𝒞_{5} with positive Gauss curvature

H_{d−1} := \frac{(−\frac{∂n}{∂u^{1}} , …, −\frac{∂n}{∂u^{d−1}} , n)}{(\frac{∂x}{∂u^{1}} , …, \frac{∂x}{∂u^{d−1}} , n)} = \frac{det(<n, \frac{∂^{2}x}{∂u^{i}∂u^{j}} >)}{det(<\frac{∂x}{∂u^{i}} , \frac{∂x}{∂u^{j}} >)}  (44)

(n inner unit normal vector of F, compare (7)). On F there exists the equiaffinely invariant (positive definite) Blaschke metric tensor field with (symmetric) coefficients

G_{ij} := \frac{(\frac{∂x}{∂u^{1}} , …, \frac{∂x}{∂u^{d−1}} , \frac{∂^{2}x}{∂u^{i}∂u^{j}})}{(det((\frac{∂x}{∂u^{1}} , …, \frac{∂x}{∂u^{d−1}} , \frac{∂^{2}x}{∂u^{i}∂u^{j}})))^{\frac{1}{d+1}}} = \frac{<n, \frac{∂^{2}x}{∂u^{i}∂u^{j}} >}{Hd−1^{\frac{1}{d+1}}} (i, j = 1, …, d − 1),  (45)

and it is possible to define there a typical (twice differentiable) affine normal vector field y by

y := \frac{1}{d − 1}Δx  (46)

(Δ Beltrami operator with respect to G_{ij}, compare (11)). For this vector field we have the affine Weingarten equations

\frac{∂y}{∂u^{i}} = −B_{i}^{j} \frac{∂x}{∂u^{j}} (i = 1, …, d − 1)  (47)

with the integrability conditions of Ricci

B_{ik} := B_{i}^{j}G_{jk} = B_{k}^{j}G_{ji} = B_{ki} (i, k = 1, …, d − 1)  (48)

such that the affine shape operator with the coefficients B_{i}^{j} (i, j = 1, …, d − 1) has d − 1 real eigenvalues called the affine principal curvatures k_{1}, …, k_{d−1} of F (compare (12)).

After these preparations we may make

Definition 4.1. An ovaloid F of class 𝒞_{5} is called affinely convex if its affine principal curvatures satisfy the conditions

k_{1} ≧ 0, …, k_{d−1} ≧ 0.  (49)

Now we have at first to study the properties of the so-called (negative) curvature image F̂ : x = −y(u^{1}, …, u^{d−1}) of F with the convex hull convF̂ in the case that F is affinely convex. We find

Proposition 4.2. Let F be an affinely convex ovaloid of class 𝒞_{5} with the curvature image F̂. Then:

i) F̂ is a part of the boundary of K̂ := convF̂. ii) The support function hK̂ of K̂ equals the \frac{1}{d+1}-th power of the Gauss curvature of the ovaloid F:

hK̂(−n) = Hd−1^{\frac{1}{d+1}} (x(n)) (n ∈ S^{d−1})  (50)

(see (44)) ^{3}.

Proof. i) It suffices to show that for any point −y_{0} = −y(u_{(0)}^{1}, …, u_{(0)}^{d−1}) of F̂ the whole set F̂ and a fortiori K̂ lies in the (convex) halfspace <n_{0}, z + y_{0}> ≧ 0 of ℝ_{d} where n_{0} := n(u_{(0)}^{1}, …, u_{(0)}^{d−1}) such that −y_{0} cannot be a point in the interior of K̂. For this purpose we choose an arbitrary point −y_{1} = −y(u_{(1)}^{1}, …, u_{(1)}^{d−1}) ∈ F̂ and join n_{1} := n(u_{(1)}^{1}, …, u_{(1)}^{d−1}) with n_{0} by the arc n(σ) = n(u^{1}(σ), …, u^{d−1}(σ)) of a great circle of S^{d−1} where σ is an (euclidean) arclength parameter (0 ≦ σ ≦ σ_{1} := ∡(n_{0}, n_{1}) ≦ π) and u^{i}(σ) are functions of σ of class 𝒞_{4}.

3In the special case H_{d−1}^{a}:= k_{1} ⋯ k_{d−1} > 0, i.e. k_{1} > 0, …, k_{d−1} > 0 this was previously proved in [4], p. 265–266.

To this arc there corresponds a curve x(σ) = x(u^{1}(σ), …, u^{d−1}(σ)) on F (with this spherical image) of class 𝒞_{4} as well as a curve −y(σ) = −y(u^{1}(σ), …, u^{d−1}(σ)) on F̂ of class 𝒞_{2}. By construction we have

n_{0} = <n(σ), n_{0}>n(σ) + <\frac{dn}{dσ}(σ), n_{0}>\frac{dn}{dσ}(σ)  (51)

with

<\frac{dn}{dσ}(σ), n_{0}> ≦ 0 (0 ≦ σ ≦ σ_{1}).  (52)

Hence by (51), (47), (45), (48), (52) and (49)

\frac{d}{dσ} <n_{0}, −y(σ) + y_{0}> = <n_{0}, −\frac{∂y}{∂u^{i}} \frac{du^{i}}{dσ} > = <\frac{dn}{dσ}, n_{0}><\frac{∂n}{∂u^{k}} \frac{du^{k}}{dσ} , B_{i}^{j} \frac{∂x}{∂u^{j}} \frac{du^{i}}{dσ} >

= −<\frac{dn}{dσ}, n_{0}>B_{i}^{j}G_{jk} Hd−1^{\frac{1}{d+1}} \frac{du^{i}}{dσ} \frac{du^{k}}{dσ} = −<\frac{dn}{dσ}, n_{0}>Hd−1^{\frac{1}{d+1}} B_{ik} \frac{du^{i}}{dσ} \frac{du^{k}}{dσ} ≧ 0

(0 ≦ σ ≦ σ_{1}) whence because of <n_{0}, −y(0) + y_{0}> = 0 after integration indeed <n_{0}, −y(σ_{1}) + y_{0}> = <n_{0}, −y_{1} + y_{0}> ≧ 0 follows. ii) In order to see (50) we choose an arbitrary direction given by the unit vector n_{0} ∈ S^{d−1} which may be considered as image of the Gauss map of a point x_{0} ∈ F with the corresponding point −y_{0} ∈ F̂. As we have seen in the proof of i) the hyperplane <n_{0}, z + y_{0}> = 0 through −y_{0} with normal direction n_{0} supports K̂ such that the (positive) distance < − n_{0}, −y_{0}> of this hyperplane from the origin yields the value of the support function hK̂ of K̂ for −n_{0} :

hK̂(−n_{0}) = <n_{0}, y_{0}>.  (53)

But from the well-known affine Gauss equations

\frac{∂^{2}x}{∂u^{i}∂u^{j}} = Γ_{ij}^{k} \frac{∂x}{∂u^{k}} + A_{ij}^{k} \frac{∂x}{∂u^{k}} + G_{ij} y (i, j = 1, …, d − 1)  (54)

where Γ_{ij}^{k} resp. A_{ij}^{k} are the Christoffel symbols for the metric (45) resp. the coefficients of the cubic form of F (i, j, k = 1, …, d − 1) we may conclude after scalar multiplication by n

<n, \frac{∂^{2}x}{∂u^{i}∂u^{j}} > = G_{ij}<n, y>.  (55)

Now the comparison of (55) and (45) yields

<n, y> = Hd−1^{\frac{1}{d+1}}  (56)

which together with (53) provides the relation (50), claimed in ii). □

One important consequence of Proposition 4.2 is the fact that the convex body K bounded by an affinely convex ovaloid F is of elliptic type in the sense of Definition 4.3. A convex body K of dimension d in ℝ_{d} is called to be of elliptic type if there exists a positive and continuous “curvature function” ρ for K on S^{d−1}, characterized by the equation

d V (K, …, K, L) = ∫_{S^{d−1}} h_{L} · ρ dω  (57)

for every convex “test body” L in ℝ_{d}, with the property

ρ^{− \frac{1}{d+1}} = h_{M}  (58)

for a suitable convex body M in ℝ_{d} (V (K, …, K, L) mixed volume, dω surface area element of S^{d−1}).

Namely if K is bounded by an ovaloid F then

ρ = H_{d−1}^{−1}  (59)

because of d V (K, …, K, L) = ∫_{F} h_{L} dℋ^{d−1} = ∫_{S^{d−1}} h_{L}H_{d−1}^{−1} dω (see (44), ℋ^{d−1} Hausdorff measure on F) for all convex bodies L (see (50) and (58)).

Now it is our aim to prove interesting results for the so-called affine surface area A(F) of an ovaloid F in ℝ_{d} of class 𝒞_{5}:

A(F) =: ∫_{F} (\frac{∂x}{∂u^{1}} , …, \frac{∂x}{∂u^{d−1}} , y) du^{1} ⋯ du^{d−1} = ∫_{F} Hd−1^{\frac{1}{d+1}} dℋ^{d−1} = ∫_{S^{d−1}} H_{d−1}^{− \frac{d}{d+1}} dω  (60)

(compare (8)) in the special case where the ovaloids are affinely convex. In this case the enclosed convex bodies are of elliptic type and the application of results of C. Petty (see [10], Theorem 3.21 and Theorems 3.12, 2.6) for bodies of elliptic type provides:

Theorem 4.4. Let A be the functional, defined by F ↦ A(F) for each affinely convex ovaloid F of class 𝒞_{5} in ℝ_{d}. Then

i) A is strictly monotone increasing, i.e.

F′ ⫅ conv F = K, F′ ≠ F ⇒ A(F′) < A(F), and  (61)

ii) A is continuous, i.e.

F = lim_{ν→∞} F_{ν} ⇒ A(F) = lim_{ν→∞} A(F_{ν}).  (62)

We have to mention that (61) is not true in general just as (62) where we have only upper semicontinuity:

F = lim_{ν→∞} F_{ν} ⇒ A(F) ≧ lim sup_{ν→∞} A(F_{ν})  (63)

(see [6], Proposition 9.2). For the sake of completeness we shall outline the Proof of Theorem 4.4. i) We use the so-called “reflected Hölder inequality”

∫_{S^{d−1}} f · g dω ≧ (∫_{S^{d−1}} f^{−d} dω)^{− \frac{1}{d}} · (∫_{S^{d−1}} g^{\frac{d}{d+1}} dω)^{\frac{d+1}{d}}  (64)

for positive and continuous functions f, g on S^{d−1} with equality if and only if

g^{\frac{d}{d+1}} = c · f^{−d} (c constant with c > 0).  (65)

Then the application of (65), (50), (64), (59) and (57) together with the monotoneity of the mixed volume yields

(∫_{s^{d−1}} h−d ; K̂ dω)^{− \frac{1}{d}} (∫_{S^{d−1}} H_{d−1}^{− \frac{d}{d+1}} dω)^{\frac{d+1}{d}} = ∫_{S^{d−1}} hK̂H_{d−1}^{−1} dω = d V (K, …, K, K̂) ≧

d V (K′, …, K′, K̂) = ∫_{S^{d−1}} hK̂H′_{d−1}^{−1} dω ≧ (∫_{S^{d−1}} h−d ; K̂ dω)^{− \frac{1}{d}} (∫_{S^{d−1}} H′_{d−1}^{− \frac{d}{d+1}} dω)^{\frac{d+1}{d}} (66)

(K′ := conv F′) whence by (60) indeed A(F) ≧ A(F′). Equality here implies equality in the second inequality of (66) whence by (65) and (50) H′_{d−1}^{−1} = c^{\frac{d+1}{d}} h−(d+1) ; K̂ = c^{\frac{d+1}{d}} H_{d−1}^{−1} and then by the equality in the first inequality of (66) c = 1 and

H′_{d−1} = H_{d−1}  (67)

follows. But an old theorem of Minkowski says that (67) implies the relation K′ = K + a (a constant with a ∈ ℝ_{d}) and thus indeed K′ = K because of K′ ⫅ K.

For this proof of i) the assumption of the affine convexity of the smaller ovaloid F′ may be omitted. Modifications of part i) of Theorem 4.4 may be found in [5]. ii) The second part of Theorem 4.4 may also be proved by reduction to the corresponding property of the mixed volume. Because of the validity of (63) we have only to show the lower semicontinuity of the functional A:

F = lim_{ν→∞} F_{ν} ⇒ A(F) ≦ lim inf_{ν→∞} A(F_{ν}).  (68)

Instead of (66) we use for this purpose the inequality

d V (K, …, K, L^{∗}) = ∫_{S^{d−1}} h_{L^{∗}}H_{d−1}^{−1} dω ≧

(∫_{S^{d−1}} h_{L^{∗}}^{−d} dω)^{− \frac{1}{d}} (∫_{S^{d−1}} H_{d−1}^{− \frac{d}{d+1}} dω)^{\frac{d+1}{d}} = (d V (L))^{− \frac{1}{d}} A(F)^{\frac{d+1}{d}} = (d κ_{d})^{− \frac{1}{d}} A(F)^{\frac{d+1}{d}} (69)

for any convex body L in ℝ_{d} whose centroid lies in the origin:

c(L) = 0  (70)

and whose volume equals the volume of the d-dimensional unit ball:

V (L) = κ_{d}  (71)

(L^{∗} polar body of L with respect to the origin). Equality holds in (69) for the body

L_{K} := (\frac{dκ_{d}}{A(F)})^{\frac{1}{d}} K̂^{∗}  (72)

since by (50)

h_{L_{K}^{∗}}^{−d} = \frac{dκ_{d}}{A(F)}h−d ; K̂ = \frac{dκ_{d}}{A(F)}H_{d−1}^{− \frac{d}{d+1}}  (73)

(see (65)), and by (73)

V (L_{K}) = \frac{1}{d} ∫_{S^{d−1}} h_{L_{K}^{∗}}^{−d} dω = \frac{κ_{d}}{A(F)} ∫_{S^{d−1}} H_{d−1}^{− \frac{d}{d+1}} dω = κ_{d}.  (74)

Hereby we also have by (50) and a theorem of Minkowski

c(L_{K}) = (\frac{dκ_{d}}{A(F)})^{\frac{1}{d}} \frac{1}{V (K̂^{∗})} ∫_{S^{d−1}} \frac{h−(d+1) ; K̂}{d + 1} (−n) dω = (⋯) ∫_{S^{d−1}}(−n)H_{d−1}^{−1} dω = 0. (75)

Now let be F = lim_{ν→∞} F_{ν} (in the Hausdorff sense) with the convex bodies K_{ν} := conv F_{ν} of elliptic type. Then we have likewise the equalities

d V (K_{ν}, …, K_{ν}, L_{K_{ν}}^{∗}) = (dκ_{d})^{− \frac{1}{d}} A(F_{ν})^{\frac{d+1}{d}}  (76)

with

V (L_{K_{ν}}) = κ_{d}, c(L_{K_{ν}}) = 0 (ν = 1, 2, …)  (77)

as for K (see (69), (73), (74) and (75)). We consider the sequence {A(F_{ν})}_{ν∈ℕ} which has a convergent subsequence {A(F_{ν′}}_{ν′∈ℕ} with

lim inf_{ν→∞} A(F_{ν}) = lim_{ν′→∞} A(F_{ν′}).  (78)

The application of Blaschke’s selection theorem to the bodies L_{K_{ν′}} which is possible because of the normalization (77) (for details see [6], proof of Lemma 6.3) yields the convergence of a subsequence of {L_{K_{ν′}}}_{ν′∈ℕ} such that we may assume, without changing the notation,

lim_{ν′→∞} L_{K_{ν′}}^{∗} =: L_{0}^{∗} with V (L_{0}) = κ_{d}, c(L_{0}) = 0.  (79)

Now we are passing to a limit for ν′ → ∞ in the equation (76) from which we may conclude using (78), (79), (69) and the continuity of the mixed volume:

(d κ_{d})^{− \frac{1}{d}} (lim inf_{ν→∞}(A(F_{ν}))^{\frac{d+1}{d}} = d V ( lim_{ν′→∞} K_{ν′}, …, lim_{ν′→∞} K_{ν′}, lim_{ν′→∞} L_{K_{ν′}}^{∗}) =

= d V (K, …, K, L_{0}^{∗}) ≧ (d κ_{d})^{− \frac{1}{d}} A(F)^{\frac{d+1}{d}}

(compare (66)) which indeed equals the claim (68) of ii). □

We end with the final

Remark 4.5. i) Proposition 4.2 and its consequence, Theorem 4.4, may be proved exactly in the same way for affinely convex ovals C in ℝ_{2} with the affine perimeter

L(C) = ∫_{C} κ^{\frac{1}{3}} dσ = ∫_{S^{1}} κ^{− \frac{2}{3}} dω  (80)

(see (8)). ii) Theorem 4.4 is also valid for affinely convex parabola polygons 𝒫 (see Definition 2.2) in ℝ_{2} with the affine perimeter (80) since their enclosed convex domains 𝒦 are of elliptic type (see Definition 4.3).

The reason of this behaviour is the fact that 𝒫 possesses the positive and continuous curvature function ρ with

ρ |_{𝒫_{l−1l}} = \frac{1}{κ^{(l−1l)}}  (81)

(see (6)) and with

<n(x_{l}), y^{(l−1l)}> = <n(x_{l}), y^{(ll+1)}> = ρ(x_{l})^{− \frac{1}{3}} (l = 1, …, k)  (82)

(see (56)) such that ρ^{− \frac{1}{3}} must be the support function of the solid convex polygon

𝒦̂ := conv {−y^{(01)}, …, −y^{(k−1k)}}  (83)

with the vertices −y^{(01)}, …, −y^{(k−1k)}, the endpoints of the negative affine normal vectors of the parabola arcs 𝒫_{01}, …, 𝒫_{k−1k}.

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Received November 7, 2000