Science Mathematics

Metric spaces, convexity and nonpositive curvature by Papadopoulos A.

By Papadopoulos A.

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In this paper, the distance between two points is called “écart”. 7 7 In fact, several basic notions in topology are due to Fréchet. In an obituary presented to the French Academy of Sciences [138], S. Mandelbrojt writes the following: “One must note that in his book written in 1914, Hausdorff, while exposing some properties of Fréchet abstract spaces, introduces a terminology, and this fact made the attribution of the corresponding notions to Hausdorff. For instance: compact sets, separable sets, semi-compact sets, metric spaces.

T/ D L. t /. Proof. Let u and u0 be two points in Œ0; L. / satisfying u Ä u0 and let t and t 0 be two points in Œa; b satisfying L. t / D u and L. t 0 / D u0 . t 0 /. t/ D L. t 0 /j Ä L. j Œt;t 0  / L. 2 Arclength as parameter which shows that is a 1-Lipschitz map. The map is increasing and surjective. 1 implies that D ı . 2. 3. Let W Œa; b ! X be a rectifiable path and let W Œ0; L. / ! 2. Then we have, for all u in Œ0; L. /, u D L. u /. Proof. Let u be a point in Œ0; L. / and let t be a point in Œa; b satisfying L.

Before that, we recall the following classical definition: x D Let E be a topological space and let x0 be a point in E. A map f W E ! x/. The map f is said to be lower semi-continuous on E if it is lower semi-continuous at every point in E. Let us note the following classical examples of lower semi-continuous maps. 2 (Lower semi-continuous maps). x is continuous at x0 in E, then f is lower semi-continuous at (i) If f W E ! R this point. (ii) The function x 7! x/ (integral value of x) is lower semi-continuous on R.

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