We prove generalizations of Meusnier's theorem and Fenchel's inequality for a class of generalized surfaces with curvature measures. Moreover, we apply them to obtain a diameter estimate.
Let (M,d) be a metric space with a fixed origin O. P. Lévy defined Brownian motion X(a); a ∈ M as 0. X(O) = 0. 1. X(a) - X(b) is subject to the Gaussian law of mean 0 and variance d(a,b). He gave an example for , the m-dimensional sphere. Let be the Gaussian random measure on , that is, 1. Y(B) is a centered Gaussian system, 2. the variance of Y(B) is equal of μ(B), where μ is the uniform measure on , 3. if B₁ ∩ B₂ = ∅ then Y(B₁) is independent of Y(B₂). 4. for , i = 1,2,..., , i ≠ j, we...
On montre que, sur une surface riemannienne compacte, le profil isopérimétrique admet un développement limité à l’ordre en . Lorsque la métrique est analytique, le profil est semi-analytique. Il existe des métriques lisses sur la -sphère dont le profil n’est pas de classe au voisinage de .