A sausage body is a unique solution for a reverse isoperimetric problem

Abstract

We consider the class of λ-concave bodies in R n+1 ; that is, convex bodies with the property that each of their boundary points supports a tangent ball of radius 1/λ that lies locally (around the boundary point) inside the body. In this class we solve a reverse isoperimetric problem: we show that the convex hull of two balls of radius 1/λ (a sausage body) is a unique volume minimizer among all λ-concave bodies of given surface area. This is in a surprising contrast to the standard isoperimetric problem for which, as it is well-known, the unique maximizer is a ball. We solve the reverse isoperimetric problem by proving a reverse quermassintegral inequality, the second main result of this paper.

A reverse isoperimetric inequality for planar ($α$, $β$)--convex bodies

“…Moreover the equality holds if and only if K is the (α, β)−egg, that is, the (α, β)−convex body whose boundary is composed by two arcs of circles of radius α joined by two arcs of circles of radius β (see figure ??). Our result together with [8] and [9]gives a complete picture of the reverse isoperimetric inequalities for convex bodies in the plane when one adds bounds on the values of the curvature.…”

“…These geometrical assumptions on local behaviour of the boundary of convex bodies have been recently used as main assumptions to prove different forms of reverse isoperimetric inequalites (cf. [9]).…”

“…] for surfaces of revolution in R 3 ). R. Chernov, K. Drach and K. Tatarko [9] studied the complement interval in terms of curvature. Indeed, they considered 0 ≤ k ≤ λ.…”

Reverse Euclidean and Gaussian isoperimetric inequalities for parallel sets with applications

“…Pan, Tang, and Wang [12] obtained a version of the reverse isoperimetric inequality for sets in R 2 by lower-bounding the perimeter in terms of the area of the set as well as the the area of the locus of its curvature centers. The one result we were able to find that holds in higher dimensions is that of Chernov, Drach, and Tatarko [13],…”

Some sharp isoperimetric-type inequalities on Riemannian manifolds

“…A counter-example is the annulus. If we allow ∂Ω to be only C 1 smooth, then a sausage body [7] is also a domain in R n with constant cut function, which is homeomorphic to but in general different from a ball. It is interesting to know under what additional assumption that Ω is a geodesic ball.…”