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Isoperimetric Inequalities in Unbounded Convex Bodies

Gian Paolo Leonardi
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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.
      Format: Paperback / softback CONTRIBUTORS: Gian Paolo Leonardi EAN: 9781470451189 COUNTRY: United States PAGES: WEIGHT: 191 g HEIGHT: 254 cm
      PUBLISHED BY: American Mathematical Society DATE PUBLISHED: 2022-06-30 CITY: GENRE: MATHEMATICS / Mathematical Analysis WIDTH: 178 cm SPINE:

      Book Themes:

      Calculus and mathematical analysis

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      Gian Paolo Leonardi, Universita di Modena e Reggio Emilia, Italy.Manuel Ritore, Universidad de Granada, Spain.Efstratios Vernadakis, Universidad de Granada, Spain.
      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.
      Format: Paperback / softback CONTRIBUTORS: Gian Paolo Leonardi EAN: 9781470451189 COUNTRY: United States PAGES: WEIGHT: 191 g HEIGHT: 254 cm
      PUBLISHED BY: American Mathematical Society DATE PUBLISHED: 2022-06-30 CITY: GENRE: MATHEMATICS / Mathematical Analysis WIDTH: 178 cm SPINE:

      Book Themes:

      Calculus and mathematical analysis

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      Gian Paolo Leonardi, Universita di Modena e Reggio Emilia, Italy.Manuel Ritore, Universidad de Granada, Spain.Efstratios Vernadakis, Universidad de Granada, Spain.

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