A sausage body is a unique solution for a reverse isoperimetric problem. (English) Zbl 1421.52008
Summary: 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 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.
MSC:
| 52A30 | Variants of convex sets (star-shaped, (\(m, n\))-convex, etc.) |
| 52A38 | Length, area, volume and convex sets (aspects of convex geometry) |
| 53A07 | Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces |
| 52A39 | Mixed volumes and related topics in convex geometry |
| 52A40 | Inequalities and extremum problems involving convexity in convex geometry |
| 52B60 | Isoperimetric problems for polytopes |
Keywords:
\(\lambda\)-concavity; \(\lambda\)-convexity; reverse isoperimetric inequality; quermassintegrals; reverse isodiametric inequalityReferences:
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