Generalización de una desigualdad geométrica de Feller

Santaló, Lluís
Using some results of integral geometry a direct and simple proof of the following inequality of Ueno-Hombu-Naito is obtained. Let D be a measurable domain of the n-dimensional space Sa of constant curvature K= k2 contained in the non-Euclidean sphere of radius a. Suppose that the intersection of D with any r-space Lr has a measure not exceeding a fixed constant ᵷ. Then the inequality (9) holds, where M is the measure of D and On—r—l, Hr.n—r—1 are given by (3) and (2) respectively. This generalizes a geometric inequality of Feller which corresponds to r=1, K=0. For the case r = l a more general inequality is given by (13) which holds for any Riemannian space not necessarily of constant curvature (F = area of a convex hyper surface which contain D) ​
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