Calculus of Variations and Geometric Measure Theory
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Archimedes, a dinner and a theorem - A divertissement on on the monotonicity of perimeter

Giorgio Stefani (Scuola Normale Superiore)

created by pagliari on 22 Apr 2018

24 apr 2018 -- 14:30   [open in google calendar]

Aula Riunioni, Dip. di Matematica, Univ. di Pisa

Abstract.

If $A$ and $B$ are two convex bodies in the Euclidean $n$-dimensional space and $A$ is contained in $B$, then the perimeter of $A$ does not exceed the perimeter of $B$. This monotonicity property of the perimeter dates back to the ancient Greek and Archimedes himself took it as a postulate in his celebrated work on the sphere and the cylinder. A few years ago, a couple of papers by M. Carozza, F. Giannetti, F. Leonetti, and A. Passarelli di Napoli established lower bounds on the difference of the perimeters of A and B in terms of their Hausdorff distance when $n=2$ and $n=3$. In this talk, after a brief introduction on the problem and the known results, I will generalise these lower bounds to any dimension $n$. Time permitting, I will show how this approach can be extended to the case of anisotropic Wulff perimeters.

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