Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 42 (2001), No. 2, 325-339.

Asymptotically Equal Generalized Distances: Induced Topologies and p-Energy of a Curve

Giuseppe De Cecco Giuliana Palmieri ^{∗}

Dipartimento di Matematica, Università di Lecce via per Arnesano – 73100, Lecce, Italy Dipartimento di Matematica, Università di Bari via Orabona, 4 – 70025, Bari, Italy

Introduction

We consider the framework of generalized metric spaces (S, σ) where S is a non-empty set and

σ : S × S → [0, +∞]

is a map such that σ(x, x) = 0. Briefly, σ is called a generalized distance. Therefore in general, σ satisfies neither symmetry nor the triangle inequality, yet it expresses the intuitive idea of a “distance”, i.e. the estimate of the “gauge” between two points.

General metric spaces were studied by Menger, Bouligand, Busemann, Pauc, Carathéodory, Blumenthal and recently by Alexandrov and Gromov ([18], [4], [5], [19], [1], [15]).

By using the weak metric structure it is possible to give a notion of convergence. If σ satisfies the separation property (σ(x, y) = 0 ⇔ x = y), i.e (S, σ) is a semimetric space, where the generalized distance is not necessarily symmetric, then it is possible to define four topologies. Moreover if σ is “continuous”, then (S, σ) is a Hausdorff topological space.

Here a particular generalized distance σ = σ_{r} (r ≥ 1) is considered, which is defined on the set S of the Lebesgue measurable subsets of R^{n}

σ_{r}(A, B) = (r ∫_{AΔB}[dist(x, ∂B)]^{r−1}dx)^{1/r}, r ≥ 1

(where AΔB is the symmetric difference of A and B). Observe that σ_{1} is the Nikodým distance.

∗ This work is partially supported by the national research projects of the MURST and CNR (Italy).

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix}2001 Heldermann Verlag

326 G. De Cecco, G. Palmieri: Asymptotically Equal Generalized Distances: …

The map σ_{r}, introduced by E. De Giorgi ([13]) as a generalization of σ_{2} and considered by Almgren-Taylor-Wang ([2]), is used in order to study the generalized minimizing motions (e.g. following the mean curvature) ([14],[20])

Other examples come from the study of the Lipschitz manifolds, which has suggested us the generalizations presented in [9].

We shall give a suitable notion of asymptotically equal generalized distances and study some of its properties. If σ is asymptotically equal to ρ and they induce a topology, then the two topologies coincide.

As in [9], if γ : [a, b] → S is a (parameterized) curve of S, it is possible to define three functionals E_{h}(σ, p) for h = 1, 2, 3 and p ≥ 1 , called p-energy of the curve γ, which generalize the usual concept. In this paper we prove that, if the generalized distances σ and ϱ are asymptotically equal, then, for h = 2, 3,

E_{h}(σ, p)(γ) = E_{h}(ϱ, p)(γ) ∀ p ≥ 1

when γ has a finite energy for some p_{0} > 1. The statement is true for every continuous curve γ if (S, σ) is a topological space.

The results (some of which are in [10]) answer a question proposed in a talk by E. De Giorgi, who in valuable discussions has drawn our attention to the problems of interactions among topology, differential geometry and calculus of variations.

1. Topology induced by a generalized distance

1.1. Let S be a set and

σ : S × S → [0, +∞]

a map such that σ(x, y) = 0 if, and only if, x = y. In Blumenthal’s language [3], (S, σ) is a semimetric space, where the generalized distance is not necessarily symmetric. For simplicity of writing we put

σ(x, y) = xy.

All that was said in [3](Ch.1, §6) for a semimetric space (with a symmetric distance) can be easily adapted to the space (S, σ) with a not symmetric distance. We give the basic concepts.

1.2. An element x ∈ S is called an L-limit (left-limit) of a sequence (x_{k}) of elements of S (briefly x_{k}⟶_{L}x or x_{k}⟶_{σ,L}x) if, and only if,

lim_{k} x_{k}x = 0,

(x_{k}x) being a sequence of non-negative real numbers. Observe that the limit may not be unique.

1.3. Let E be a subset of S. An element x ∈ S is called an L-accumulation point of E provided that, for each positive number ε, there is a point y ∈ E such that 0 < yx < ε.

The subset E is L-closed if it contains each one of its accumulation points. E is L-open provided its complement C(E) is L-closed. The family of all L-open sets, defined above, is closed under arbitrary unions and finite intersections; therefore, it forms a topology, named T_{L}(σ) or briefly T_{L}.

1.4. Let x, y be elements of S. If for any sequence (y_{k}) of elements of S,

(y_{k})⟶_{L}y ⇒ (y_{k}x) → yx,

then the distance function σ is said to be L-continuous at y, x; it is continuous in S provided it is continuous at each pair of points of S.

1.5. If x ∈ S and ε is a positive number, the subset

B_{L}(x; ε) = {y ∈ S; yx < ε}

is called the L-spherical neighborhood of x with radius ε. Observe that a spherical neighborhood need not be open, nevertheless if σ is an L-continuous distance function, then, for all x ∈ S and ε > 0, the sets B_{L}(x; ε) are L-open and they form a base for the topology T_{L}(σ).

1.6. All that was said can be repeated interchanging the roles of left and right. An element x ∈ S is called an R-limit of a sequence (x_{k}) of elements of S if, and only if,

lim_{k} xx_{k} = 0.

An element x ∈ S is called an R-accumulation point of E provided that for every positive number ε, there is a point y ∈ E such that 0 < xy < ε. The family of all R-open sets forms a topology on S, named T_{R}. If σ is R-continuous in x, y, then the sets

B_{R}(x; ε) = {y ∈ S; xy < ε}

are R-open.

1.7. Example. Let S = R and

σ(x, y) = \{ \begin{matrix}y − x, & x < y, \\ 0, & x ≥ y. \end{matrix}

The map σ is not symmetric, but satisfies the triangle inequality; moreover it is Rcontinuous. The spherical neighborhoods are the sets B_{R}(x; ε) = (−∞, x + ε], which generate on R the topology of upper semicontinuity. Analogously, B_{L}(x; ε) = [x − ε, +∞) generate on R the topology of lower semicontinuity.

1.8. An element x ∈ S is called a w-limit (weak-limit) of a sequence (x_{k}) of elements of S if, and only if,

min{lim_{k} x_{k}x, lim_{k} xx_{k}} = 0.

An element x ∈ S is called a w-accumulation point of E provided that, for every positive number ε, there is a point y ∈ E such that 0 < min{yx, xy} < ε. The family of all w-open sets forms the topology T_{w}.

1.9. Analogously, an element x ∈ S is called an s-limit (strong-limit) of a sequence (x_{k}) of elements of S if, and only if,

max{lim_{k} x_{k}x, lim_{k} xx_{k}} = 0.

An element x ∈ S is called an s-accumulation point of E provided that, for every positive number ε, there is a point y ∈ E such that 0 < max{yx, xy} < ε. The family of all s-open sets forms the topology T_{s}.

The topology T_{w} is the weakest of the four topologies, while T_{s} is the strongest. In general the four topologies may be distinct even if σ is continuous (with respect to T_{w} and hence with respect to the others).

We summarize the results in the following theorem, which was previously known if the distance function is symmetric:

1.10. Theorem. On a semimetric space (S, σ), where σ is a generalized (not necessarily) symmetric distance, the topologies T_{h} (h = L, R, w, s) can be defined. If σ is continuous with respect to T_{h}, then (S, T_{h}) is a Hausdorff space and the balls B_{h}(x; ε) form a base for the neighborhoods. Moreover, if σ is continuous with respect to the weak topology T_{w}, then S is Hausdorff also with respect to the other topologies T_{h} (h = L, R, s).

Examples

1.11. Let S = R and

σ(x, y) = \{ \begin{matrix}y − x, & y ≥ x, \\ 1, & y < x. \end{matrix}

The w-topology is the Euclidean one, the s-topology is the discrete one, the spherical neighborhoods of T_{L} and T_{R} are respectively

B_{L}(x; ε) = [x, x + ε), B_{R}(x; ε) = (x − ε, x].

Because σ(x_{k}, x) → 0 if, and only if, σ(x, x_{k}) → 0, the four topologies are continuous.

The following examples have suggested us to consider general metric spaces ([9], [19]).

1.12. If (M, F) (resp. (M, g)) is a Finsler (smooth) manifold (resp. a Riemann manifold), the function σ : M × M → R^{+} defined in a chart (U, Φ) by

σ̃(ξ, η) = F (ξ, η − ξ)  or  σ̃(ξ, η) = [∑_{h,k} g_{h,k}(ξ)(η_{h} − ξ_{h})(η_{k} − ξ_{k})]^{1/2} ,

induces on M the generalized distance σ(x, y) = σ̃(Φ(x), Φ(y)), which satisfies neither symmetry nor the triangle inequality. Thus (M, σ) becomes a general metric space, hence a topological space, because σ is continuous (nay smooth); moreover the previous topologies coincide.

1.13. The spaces (S, T_{h}) in general are not metric, however the following statement holds:

1.14. Theorem. Let σ and ρ be two generalized distances on S. A necessary and sufficient condition in order that, for h = L, R, w, s, the topology T_{h}(σ) coincides with T (ρ) is that

x_{k}⟶_{σ,h}x ⇔ x_{k}⟶_{ρ,h}x

Proof. We prove the theorem for h = L; in the other cases we can proceed in an analogous manner.

Let ρ(x_{k}, x) → 0 be with x_{k} ≠ x and lim sup_{k} σ(x_{k}, x) = a > 0, then it is possible to extract a subsequence of (x_{k}), denoted (y_{n}), such that

σ(y_{n}, x) > 0, lim_{n} σ(y_{n}, x) = a, (lim_{n} ρ(y_{n}, x) = 0).

If C denotes the closure of the set {y_{n}; n ∈ N} with respect to T_{L}(σ), then C is not closed with respect to T_{L}(ρ), provided x ∉ C. The statement of the theorem is obtained by interchanging the roles of σ and ρ. □

It follows easily that

1.15. Theorem. Let σ and ρ be two generalized distances on S. A sufficient condition in order that, for h = L, R, w, s, the topology T_{h}(σ) coincides with T_{h}(ρ) is that

lim sup_{x_{k}⟶_{σ,L}x} \frac{σ(x_{k}, x)}{ρ(x_{k}, x)} < +∞, lim sup_{x_{k}⟶_{ρ,L}x} \frac{ρ(x_{k}, x)}{σ(x_{k}, x)} < +∞.

Analogous conditions, mutatis mutandis, hold for the topologies T_{R}, T_{w}, T_{s}.

1.16. Two generalized distances σ and ρ are called equivalent if

x_{k}⟶_{ρ,w}x, y_{k}⟶_{ρ,w}y ⇒ lim sup_{k} \frac{σ(x_{k}, y_{k})}{ρ(x_{k}, y_{k})} < +∞

and

x_{k}⟶_{σ,w}x, y_{k}⟶_{σ,w}y ⇒ lim sup_{k} \frac{ρ(x_{k}, y_{k})}{σ(x_{k}, y_{k})} < +∞.

Naturally the previous conditions are satisfied if two real numbers a, b exist such that

aσ(x, y) ≤ ρ(x, y) ≤ bσ(x, y) ∀x, y ∈ S

which is the usual condition in metric spaces. From Theorem 1.15 we have

1.17. Theorem. If σ and ρ are equivalent, then T_{h}(σ) = T_{h}(ρ) for h = L, R, w, s.

2. A remarkable example

2.1. Let S̃ be the set of the Lebesgue measurable subsets of R^{n} and, for all A, B ∈ S̃, define

σ_{r}(A, B) = (r ∫_{AΔB}[dist(x, ∂B)]^{r−1}dx)^{1/r} (r ≥ 1)

(where AΔB is the symmetric difference of A and B).

Clearly, (S̃, σ_{r}) is a general metric space. When we identify two sets A and B such that |AΔB| = 0, then σ_{1} is the Nikodým distance, while σ_{r} (r > 1) is not a distance in the usual sense, namely σ_{r}(A, B) ≠ σ_{r}(B, A).

In order to avoid pathological behavior, it is convenient to restrict S̃ to more meaningful subsets,

S = {X ⊂ R^{n}; X convex and bounded}

or

K = {X ⊂ R^{n}; X a convex body}.

Now, for all A, B ∈ S, with the above identification,

σ_{r}(A, B) = 0 ⇒ A = B.

In [21] the following statements are proved:

2.2. Theorem. Let A, B ∈ S with |A| ≠ 0, |B| ≠ 0. If (A_{k}), (B_{k}) are sequences in S and (A_{k}) → A, (B_{k}) → B in the topology of σ_{1}, then

σ_{r}(A_{k}, B_{k}) → σ_{r}(A, B).

2.3. Theorem. Let (A_{k}) be a sequence in S and A ∈ S. Then

σ_{r}(A_{k}, A) → 0 ⇔ σ_{1}(A_{k}, A) → 0,

hence

σ_{r}(A_{k}, A) → 0 ⇔ σ_{r}(A, A_{k}) → 0.

Hence the generalized distance σ_{r} is continuous and the four topologies T_{h}(σ_{r}) are equal. Moreover, by Theorem 2.3, these topologies coincide with the one induced by σ_{1}, i.e. the Nikodým topology, which is the topology induced also by the Hausdorff distance ([16]).

3. Asymptotically equal distances

3.1. Let σ and ϱ be two generalized distances. We say that σ is asymptotically equal to ϱ at x ∈ S if, and only if,

x_{k}⟶_{σ}x, y_{k}⟶_{σ}x ⇒ lim_{k} \frac{σ(x_{k}, y_{k})}{ϱ(x_{k}, y_{k})} = 1.

If σ is asymptotically equal to ϱ at all points x ∈ S, then we write σ ∼ ϱ. In general σ ∼ ϱ does not imply ϱ ∼ σ, as is shown by the following example.

3.2. Example. Let S = R and

ϱ(x, y) = | sin σ(x, y)|,

where σ might be a distance in the usual sense, in particular σ(x, y) = |x − y|. Now σ ∼ ϱ, but if x̄, ȳ ∈ S are two points s.t. σ(x̄, ȳ) = mπ (m ∈ N \ {0}) then ϱ(x̄, ȳ) = 0, hence ϱ is not asymptotically equal to σ.

We remark that, for example, the σ-closure of (x_{k}), where x_{k} = 1/k is {x_{k}; k ∈ N} ∪ {0}, while the ρ-closure is {x; x = mπ, m ∈ N}. □

Observe that if σ ∼ ϱ and a real number a > 0 exists such that

aσ(x, y) ≤ ϱ(x, y),

then ϱ ∼ σ.

3.3. Theorem. Let σ, ρ be two generalized distances on the set S. If σ ∼ ρ and ρ ∼ σ, then σ and ρ induce the same topology on S.

Proof. If x is an L-accumulation point of a set E ⊂ S, then a sequence (x_{k}), with x_{k} ∈ E \ {x}, exists such that σ(x_{k}, x) → 0. By definition, one has ρ(x_{k}, x) → 0 too (and vice versa reversing the roles of σ and ρ), also the L-accumulation points with respect to the topology induced by σ coincide with the L-accumulation points with respect to the topology induced by ρ. Analogous conclusions hold in the other cases. □

Observe that σ and ρ may induce the same topology, without being asymptotically equal (for example σ_{1} and σ_{r}).

The LIP case

3.4. Let (M, δ) be a LIP manifold, where δ is a distance locally equivalent to a Euclidean one. If (U, Φ) is a chart at the point x ∈ M, ξ = Φ(x), v is a vector of V = Φ(U) ⊂ R^{n}, we consider the “directional derivative” of δ at the point ξ,

φ(ξ, v) = lim sup_{t→0^{+}} \frac{δ(Φ^{−1}(ξ), Φ^{−1}(ξ + tv))}{t}.

For almost all ξ ∈ V there exists the limit and the function φ(ξ, ·) is a norm that depends on ξ and which is locally equivalent to the Euclidean norm ([7]). Then

d̃(ξ, η) = φ(ξ, η − ξ)

is a generalized distance on R^{n}, not continuous, which satisfies neither symmetry nor the triangle inequality.

3.5. Theorem. Let (M, δ) be a LIP manifold, where δ is a distance locally equivalent to a Euclidean one. If d̃(ξ, η) = φ(ξ, η − ξ) is the generalized distance induced on the chart, then δ is a.e. asymptotically equal to d, where

d(x, y) = d̃(ξ, η), x = Φ^{−1}(ξ), y = Φ^{−1}(η).

Proof. If δ(Φ^{−1}(ξ), Φ^{−1}(η)) = δ̃(ξ, η), by (3.4) one has for ξ ≠ η,

\frac{δ̃(ξ, η)}{d̃(ξ, η)} = \frac{δ̃(ξ, η)}{φ(ξ, η − ξ)} = \frac{δ̃(ξ, ξ + ∥η − ξ∥\frac{η−ξ}{∥η−ξ∥})}{φ(ξ, \frac{η−ξ}{∥η−ξ∥})∥η − ξ∥}.

From every sequence (η_{k}) such that η_{k} → ξ, it is possible to extract a subsequence (denoted again η_{k}) such that

\frac{η_{k} − ξ}{∥η_{k} − ξ∥} → v, ∥v∥ = 1.

Then, for almost all ξ, one has δ ∼ d because

lim_{k→+∞} \frac{δ(ξ, η_{k})}{d(ξ, η_{k})} = lim_{k→+∞} \frac{δ̃(ξ, η_{k})}{d̃(ξ, η_{k})} = \frac{φ(ξ, v)}{φ(ξ, v)} = 1.  □

3.6. Theorem. Let (M, δ) be a LIP manifold, where δ is a distance locally equivalent to a Euclidean one. If ρ is a distance (on M) asymptotically equal to δ, then, for almost all ξ

φ^{δ}(ξ, v) = φ^{ρ}(ξ, v)

where φ^{δ} (resp. φ^{ρ}) is the “directional derivative” of δ (resp. ρ).

Proof. At the points where φ^{δ} and φ^{ρ} exist and, by the definition of asymptoticity, the relation

lim_{t→0} \frac{δ(Φ^{−1}(ξ), Φ^{−1}(ξ + tv))}{t} \frac{t}{ρ(Φ^{−1}(ξ), Φ^{−1}(ξ + tv))} = 1.

holds, whence the conclusion. □

It follows in particular that

3.7. Theorem. Let M be a metric space with respect to two asymptotically equal distances δ and ρ. Moreover let A be an open subset of R^{n} and f : A → M a LIP map. If E ⊂ f(A) is H_{δ}^{n}-measurable (where H_{δ}^{n} is the Hausdorff measure induced by δ) then E is H_{ρ}^{n}-measurable and

H_{δ}^{n}(E) = H_{ρ}^{n}(E).

It is sufficient to recall a representation theorem of type “area” ([17], [11,(3.7)]). Because for the length of a curve γ constructed from the distance σ one has [6]

L(γ; σ) = ∫_{a}^{b} φ^{σ}(γ, γ̇)dt

it follows that:

3.8. Theorem. Let M be a metric space with respect to two asymptotically equal distances δ and ρ. If γ is a curve of M, then

L(γ; δ) = L(γ; ρ).

3.9. Example. Let (M, g) be a LIP Riemannian manifold embedded in (R^{n}, d), where d is the standard distance. If δ^{g} is the intrinsic distance induced on M by g ([6],[7]), then δ^{g} ∼ d a.e. on M. Namely by Theorem [7,(6.2)], for almost all y

lim_{x→y} \frac{δ^{g}(x, y)}{d(x, y)} = 1.

We recall that it is possible to have a LIP manifold (M, g) with φ(ξ, ˙ ) a norm, that is not derived from an inner product. Hence

3.10. Theorem. [7,(6.3)] Given a LIP Riemannian manifold (M, g), in general it is not possible to find a number m ∈ N such that (M, g) is isometric to a LIP submanifold of (R^{n}, nat).

4. p-Energy of a curve

4.1. As in [9], if γ : [a, b] → S is a (parameterized) curve of S, a ≤ t′ < t′′ ≤ b and T = {t′ = t_{0} < t_{1} < ... < t_{n+1} = t′′} is a decomposition of [t′, t′′], we define for p ≥ 1, p ∈ R, the following functionals, called p-energies of the curve γ,

E_{1}(σ, p)(γ; t′, t′′) = sup_{T}\{∑_{i=0}^{n} \frac{σ(γ(t_{i}), γ(t_{i+1}))^{p}}{(t_{i+1} − t_{i})^{p−1}} \};

E_{2}(σ, p)(γ; t′, t′′) = inf_{T}{∑_{i=0}^{n} E_{1}(σ, p)(γ; t_{i}, t_{i+1})};

E_{3}(σ, p)(γ; t′, t′′) = ∫_{∗}_{t′}^{t′′}(lim sup_{h→0} \frac{σ(γ(t), γ(t + h))^{p}}{h^{p}} )dt;

(where this latter integral is meant as a Lebesgue upper integral).

The functional E_{1} can be considered as the total p-variation of γ, with respect to the function σ. If σ is a distance and p = 1 we have the usual concept of length of a curve, for p = 2 we have the extension of the concept of energy to curves, that need not be smooth. In the general case,

E_{1} ≥ E_{2} ≥ E_{3}

and there exist examples for which strict inequalities hold.

4.2. We say that γ satisfies the finite energy condition for E_{h} if some p_{0} > 1 exists such that E_{h}(σ, p_{0})(γ) < +∞.

4.3. Theorem. [9] If σ satisfies the triangle inequality (on γ(I)), then

E_{1}(σ, p)(γ) = E_{2}(σ, p)(γ) ∀p ≥ 1.

Moreover if γ satisfies the finite energy condition, then

E_{1}(σ, p)(γ) = E_{2}(σ, p)(γ) = E_{3}(σ, p)(γ) = E(σ, p)(γ) ∀p ≥ 1.

If S is a LIP (topological) manifold M and σ a distance δ locally equivalent to the Euclidean one, then

E(δ, p)(γ; a, b) = ∫_{a}^{b} φ(γ, γ̇)^{p}dt.

where φ is the “derivative” of δ (see (3.4)).

In particular, if S is a LIP Finslerian manifold of class C^{1} and δ = δ^{F} is the intrinsic distance induced by a continuous norm F, then φ = F. We recall ([7]) that if F is a generic Finslerian structure, then in general φ ≠ F, but ϱ^{φ} = ϱ^{F}.

Examples

4.4. We consider Example 3.2, where S = R and

γ(t) = \{ \begin{matrix}x̄, & [a, b] ∩ Q, \\ ȳ, & [a, b] ∩ R − Q \end{matrix} .

Then σ(γ(t), γ(t + h)) = 0, mπ, while ρ(γ(t), γ(t + h)) = 0. One easily sees that

E_{3}(σ, p)(γ) = +∞, E_{3}(ϱ, p)(γ) = 0.

It follows that one may have σ ∼ ρ but E_{3}(σ, p)(γ) ≠ E_{3}(ϱ, p)(γ).

4.5. Even if σ ∼ ρ and ρ ∼ σ, this does not imply that the energies are equal. Indeed, let S = R and

σ(x, y) = |x − y|, ϱ(x, y) = e^{σ(x,y)}σ(x, y),

G. De Cecco, G. Palmieri: Asymptotically Equal Generalized Distances: … 335

then σ ∼ ϱ, ϱ ∼ σ, but

E_{1}(ϱ, p)(γ) = e^{p(b−a)}(b − a) > (b − a) = E_{1}(σ, p)(γ).

5. The main theorems If M is a LIP (topological) manifold with σ and ρ distances (locally equivalent to a Euclidean one and) asymptotically equal, then, for every curve γ of M,

E_{h}(σ, p)(γ) = E_{h}(ρ, p)(γ) h = 1, 2, 3; p ≥ 1.

Now we shall study under what conditions the energies are equal in the case that the generalized distances are asymptotically equal on a set M. 5.1. Theorem. Let σ and ρ be generalized distances and σ ∼ ρ. If γ is a curve of S such that E_{3}(σ, 1)(γ) < +∞, then E_{3}(ρ, 1)(γ) < +∞ too and

E_{3}(σ, p)(γ) = E_{3}(ρ, p)(γ) ∀p ≥ 1.

Proof. The condition E_{3}(σ, 1)(γ) < +∞ gives, for almost all t ∈ [a, b],

lim sup_{h→0^{+}} \frac{σ(γ(t), γ(t + h))}{h} ∈ R ⇒ lim_{h→0} σ(γ(t), γ(t + h)) = 0.

Because σ ∼ ρ, for every sequence (h_{n}) (with h_{n} ≥ 0) convergent to 0,

lim_{n} \frac{σ(γ(t), γ(t + h_{n}))}{h_{n}} · \frac{h_{n}}{ρ(γ(t), γ(t + h_{n}))} = 1

holds and hence, if we choose a sequence (which we again indicate (h_{n})) such that

lim_{n} \frac{σ(γ(t), γ(t + h_{n}))}{h_{n}} = lim sup_{n} \frac{σ(γ(t), γ(t + h_{n}))}{h_{n}} = ψ(γ(t)),

it follows that

lim sup_{n} \frac{ρ(γ(t), γ(t + h_{n}))}{h_{n}} ≥ lim_{n} \frac{ρ(γ(t), γ(t + h_{n}))}{h_{n}} = ψ(γ(t)).

We choose (h_{n}) such that

lim_{n} \frac{ρ(γ(t), γ(t + h_{n}))}{h_{n}} = lim sup_{n} \frac{ρ(γ(t), γ(t + h_{n}))}{h_{n}};

then we obtain the opposite inequality, whence the conclusion. □

5.2. Theorem. Let σ and ρ be generalized distances and σ ∼ ρ. If γ is a curve of S such that E_{2}(σ, p_{0})(γ) < +∞ for some p_{0} > 1, then E_{2}(ρ, p_{0})(γ) < +∞ too and

E_{2}(σ, p)(γ) = E_{2}(ρ, p)(γ) ∀p ≥ 1.

Proof. First we remark that E_{1}(σ, p_{0})(γ) < +∞, for some p_{0} > 1, implies for t < τ

(5.3)  lim_{t,τ→t^{∗}} σ(γ(t), γ(τ)) = 0,

i.e. the continuity of σ(γ(t), γ(τ)) at the point (t^{∗}, t^{∗}) of the diagonal; but in general the continuity of σ(γ(t), γ(τ)), as a function of (t, τ) does not follow.

i) If E_{1}(σ, p_{0})(γ) < +∞ for some p_{0} > 1, then one proves that, ∀ε > 0 a δ_{ε} exists such that

(1 − ε) < \frac{σ(γ(t), γ(τ))}{τ(γ(t), γ(τ))} < (1 + ε), 0 < τ − t < δ_{ε}.

Indeed, suppose ab absurdo that, ∀n the points t_{n}, τ_{n} ∈ [a, b] exist s.t.

(5.4)  a ≤ t_{n} < τ_{n} ≤ b, τ_{n} − t_{n} < \frac{1}{n}, |\frac{σ(γ(t_{n}, γ(τ_{n}))}{ρ(γ(t_{n}), γ(τ_{n}))} − 1| ≥ ε.

It is possible to choose subsequences, which we again call (t_{n}), (τ_{n}), convergent to a point t^{∗}. Then by (5.3) ∀n one has γ(t_{n})⟶_{σ}γ(t^{∗}), γ(τ_{n})⟶_{σ}γ(t^{∗}), and by the assumptions lim_{n} σ(γ(t_{n}, γ(τ_{n}))/ρ(γ(t_{n}), γ(τ_{n})) = 1, which contradicts (5.4).

Let T̄ be a partition of [a, b] with width smaller than δ_{ε}. Then

(1 − ε)^{p} \frac{ρ(γ(t_{n}), γ(t_{n+1}))^{p}}{(t_{n+1} − t_{n})^{p−1}} ≤ \frac{σ(γ(t_{n}), γ(t_{n+1}))^{p}}{(t_{n+1} − t_{n})^{p−1}} ≤ \frac{ρ(γ(t_{n}), γ(t_{n+1}))^{p}}{(t_{n+1} − t_{n})^{p−1}} (1 + ε)^{p},

whence

(1 − ε)^{p} ∑_{i=0}^{n} E_{1}(ρ, p)(γ; t_{i}, t_{i+1}) ≤ ∑_{i=0}^{n} E_{1}(σ, p)(γ; t_{i}, t_{i+1}) ≤

≤ (1 + ε)^{p} ∑_{i=0}^{n} E_{1}(ρ, p)(γ; t_{i}, t_{i+1}).

Since

inf_{T ⊃T̄}{∑_{i=0}^{n} E_{1}(σ, p)(γ; t_{i}, t_{i+1})} = inf_{T}{∑_{i=0}^{n} E_{1}(σ, p)(γ; t_{i}, t_{i+1})} = E_{2}(σ, p)(γ; t′, t′′),

by the arbitrariness of ε the assertion of the theorem follows. (ii) By the definition of E_{2} and because of the assumptions, a partition of [a, b] exists such that E_{1}(σ, p_{0})(γ; t_{i}, t_{i+1}) < +∞. Then by (i)

E_{2}(σ, p_{0})(γ; t_{i}, t_{i+1}) = E_{2}(ρ, p_{0})(γ; t_{i}, t_{i+1}),

G. De Cecco, G. Palmieri: Asymptotically Equal Generalized Distances: … 337

from which the conclusion follows provided E_{2} is an additive function. □ Remarks 5.5. The result of the Theorem 5.2 is not true for E_{1}, as the Example 4.5 shows. 5.6. The condition E_{2}(σ, 1)(γ) < +∞ does not imply the equality E_{2}(σ, p)(γ) = E_{2}(ϱ, p)(γ) even for finite energies. For example, if

γ(t) = \{ \begin{matrix}x̄, & a ≤ t ≤ c, \\ ȳ, & c ≤ t ≤ b, \end{matrix}

and σ(x̄, ȳ) ≠ ϱ(x̄, ȳ), we have

E_{2}(σ, 1)(γ) = σ(x̄, ȳ) ≠ ϱ(x̄, ȳ) = E_{2}(ϱ, 1)(γ).

5.7. For the equality in 5.2 the condition E_{3}(σ, 1)(γ) < +∞ is essential, as Example 4.4 shows. 5.8. The conditions

E_{3}(σ, 1)(γ) < +∞, E_{2}(σ, p_{0})(γ) < +∞, p_{0} > 1,

can be replaced by

lim_{t_{n}→t^{−}} σ(γ(t_{n}), γ(t)) = 0, lim_{t_{n}→t^{+}} σ(γ(t), γ(t_{n})) = 0.

5.9. If E_{2}(σ, p)(γ) = +∞ for all p > 1, then the result of the Theorem 5.2 is true if

\frac{ρ(γ(t), γ(τ))}{σ(γ(t), γ(τ))} ≥ c, ∀t, τ ∈ [a, b].

For example, if for t_{n} → t,

lim sup_{n} σ(γ(t), γ(t_{n})) = l > 0,

then

lim sup_{n} ρ(γ(t), γ(t_{n})) ≥ cl > 0,

and hence

E_{2}(σ, p)(γ) = E_{2}(ρ, p)(γ) = +∞, ∀p > 1.

From the remark in 5.7 it follows that: 5.10. Theorem. Let S be a topological space and σ a continuous map. If σ ∼ ρ, then, for every continuous curve γ,

E_{h}(σ, p)(γ) = E_{h}(ρ, p)(γ) ∀p ≥ 1, h = 2, 3.

Remarks

5.11. For h = 1, the theorem is not true as shown in Example 4.5.

5.12. The Nikodým distance σ_{1} and the generalized distance σ_{r} (introduced in Section 2) induce the same topology, but σ_{1} is not asymptoyically equal to σ_{r}, because E_{h}(σ_{1}, p)(γ) ≠ E_{h}(σ_{r}, p)(γ) (see [9], §5).

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Received February 11, 2000