Isoperimetric Inequalities in Unbounded Convex Bodies
Isoperimetric Inequalities in Unbounded Convex Bodies
Isoperimetric Inequalities in Unbounded Convex Bodies
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Isoperimetric Inequalities in Unbounded Convex Bodies

Leonardi, Gian Paolo

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Paperback / softback
30 June 2022
86 Pages
$197.00
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We consider the problem of minimizing the relative perimeter under a volume constraint in an unbounded convex body C ? Rn, without assuming any further regularity on the boundary of C. Motivated by an example of an unbounded convex body with null isoperimetric profile, we introduce the concept of unbounded convex body with uniform geometry. We then provide a handy characterization of the uniform geometry property and, by exploiting the notion of asymptotic cylinder of C, we prove existence of isoperimetric regions in a generalized sense. By an approximation argument we show the strict concavity of the isoperimetric profile and, consequently, the connectedness of generalized isoperimetric regions. We also focus on the cases of small as well as of large volumes; in particular we show existence of isoperimetric regions with sufficiently large volumes, for special classes of unbounded convex bodies. We finally address some questions about isoperimetric rigidity and analyze the asymptotic behavior of the isoperimetric profile in connection with the notion of isoperimetric dimension.
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Isoperimetric Inequalities in Unbounded Convex Bodies

$197.00

Description

We consider the problem of minimizing the relative perimeter under a volume constraint in an unbounded convex body C ? Rn, without assuming any further regularity on the boundary of C. Motivated by an example of an unbounded convex body with null isoperimetric profile, we introduce the concept of unbounded convex body with uniform geometry. We then provide a handy characterization of the uniform geometry property and, by exploiting the notion of asymptotic cylinder of C, we prove existence of isoperimetric regions in a generalized sense. By an approximation argument we show the strict concavity of the isoperimetric profile and, consequently, the connectedness of generalized isoperimetric regions. We also focus on the cases of small as well as of large volumes; in particular we show existence of isoperimetric regions with sufficiently large volumes, for special classes of unbounded convex bodies. We finally address some questions about isoperimetric rigidity and analyze the asymptotic behavior of the isoperimetric profile in connection with the notion of isoperimetric dimension.

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