Isoperimetric inequalities in Euclidean convex bodies

Isoperimetric inequalities in convex cylinders and cylindrically bounded convex bodies

“…Next we consider small and large volumes. For small volumes, following Ritoré and Vernadakis [17], we show in Theorem 3.7 that the isoperimetric profile of a convex cylinder for small volumes is asymptotic to the one of its narrowest tangent cone. As a consequence, we completely characterize the isoperimetric regions of small volumes in a convex prism, i.e, a cylinder P × q based on a convex polytope P ⊂ m .…”

“…Proof. Since the quotient of C by its isometry group is compact, the proof is reduced to that of [17,Thm. 4.12].…”

“…Let us consider now the isoperimetric profile for small volumes. The following is inspired by [17,Theorem 6.6], although we have simplified the proof. Theorem 3.7.…”

“…To accomplish this we consider λ i → ∞ so that the isoperimetric regions λ i E ⊂ λ i C have volume 1. Then we argue exactly as in [17,Theorem 6.6]. We first produce an elimination Lemma as in [17, Theorem 5.5], with ǫ > 0 independent of λ i , that yields a perimeter lower density bound [17, Corollary 5.8] independent of λ i .…”

“…No assumption on the regularity of ∂ C will be made. Existence of isoperimetric regions is obtained in Proposition 3.2 following the scheme of proof by Galli and Ritoré [4], which essentially needs a uniform local relative isoperimetric inequality [17], a doubling property on K × q given in Lemma 3.1, an upper bound for the isoperimetric profile of C given in (2.6), and a well-known deformation controlling the perimeter in terms of the volume. A proof of existence of isoperimetric regions in Riemannian manifolds with compact quotient under their isometry groups was previously given by Morgan [13].…”