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

@article{Chernov2019ASB,
  title={A sausage body is a unique solution for a reverse isoperimetric problem},
  author={Roman Chernov and Kostiantyn Drach and Kateryna Tatarko},
  journal={Advances in Mathematics},
  year={2019}
}
We consider the class of $\lambda$-concave bodies in $\mathbb R^{n+1}$; that is, convex bodies with the property that each of their boundary points supports a tangent ball of radius $1/\lambda$ 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 $1/\lambda$ (a sausage body) is a unique volume minimizer among all $\lambda$-concave bodies of given surface area. This is in a… Expand

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References

SHOWING 1-10 OF 47 REFERENCES
Reverse Bonnesen style inequalities in a surface $$\mathbb{X}_\varepsilon ^2$$ of constant curvature
We investigate the isoperimetric deficit upper bound, that is, the reverse Bonnesen style inequality for the convex domain in a surface $$\mathbb{X}_\varepsilon ^2$$ of constant curvature ɛ via theExpand
Volume Ratios and a Reverse Isoperimetric Inequality
It is shown that if $C$ is an $n$-dimensional convex body then there is an affine image $\widetilde C$ of $C$ for which $${|\partial \widetilde C|\over |\widetilde C|^{n-1\over n}}$$ is no largerExpand
A Reverse Isoperimetric Inequality for Convex Plane Curves ∗
In this note we present a reverse isoperimetric inequality for closed convex curves, which states that if γ is a closed strictly convex plane curve with length L and enclosing an area A, then oneExpand
Relation between area and volume for λ-convex sets in Hadamard manifolds☆
Abstract It is known that for a sequence {Ω t } of convex sets expanding over the whole hyperbolic space H n+1 the limit of the quotient vol (Ω t )/ vol (∂Ω t ) is less or equal than 1/n , andExpand
The Isoperimetric Problem
TLDR
Virgil tells us that Aeneas, on his quest to found Rome, is shipwrecked and blown ashore at Carthage and Dido falls in love with him, but he does not return her love, and she kills herself. Expand
Convex sets in Hadamard manifolds
Abstract We give sharp upper estimates for the difference circumradius minus inradius and for the angle between the radial vector (respect to the center of an inball) and the normal to the boundaryExpand
A Reverse Isoperimetric Inequality, Stability and Extremal Theorems for Plane Curves with Bounded Curvature
Stability and Extremal Theorems for Plane Curves with Bounded Curvature Ralph Howard and Andrejs Treibergs In this note we dis uss some elementary theorems about the relation between area and lengthExpand
Inequalities for hyperconvex sets
Abstract An r-hyperconvex body is a set in the d-dimensional Euclidean space 𝔼d that is the intersection of a family of closed balls of radius r. We prove the analogue of the classicalExpand
Ball-Polyhedra
TLDR
This work finds analogues of several results on convex polyhedral sets for ball-polyhedra by studying two notions of spindle convexity and bodies obtained as intersections of finitely many balls of the same radius. Expand
Closeness to spheres of hypersurfaces with normal curvature bounded below
For a Riemannian manifold M{sup n+1} and a compact domain Ω⊂ M{sup n+1} bounded by a hypersurface ∂Ω with normal curvature bounded below, estimates are obtained in terms of the distance from O to ∂ΩExpand
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