04593nam 22006013 450 991096597460332120251116135918.097814704702031470470209(MiAaPQ)EBC6939729(Au-PeEL)EBL6939729(CKB)21420569700041(OCoLC)1321796883(EXLCZ)992142056970004120220327d2022 uy 0engurcnu||||||||txtrdacontentcrdamediacrrdacarrierIsoperimetric Inequalities in Unbounded Convex Bodies1st ed.Providence :American Mathematical Society,2022.©2022.1 online resource (100 pages)Memoirs of the American Mathematical Society ;v.276Print version: Leonardi, Gian Paolo Isoperimetric Inequalities in Unbounded Convex Bodies Providence : American Mathematical Society,c2022 9781470451189 Cover -- Title page -- List of symbols -- Chapter 1. Introduction -- 1.1. Historical background -- 1.2. Outline of contents -- Chapter 2. Convex bodies and finite perimeter sets -- 2.1. Convex bodies and local convergence in Hausdorff distance -- 2.2. Finite perimeter sets and isoperimetric profile -- Chapter 3. Unbounded convex bodies of uniform geometry -- 3.1. Asymptotic cylinders -- 3.2. Convex bodies of uniform geometry -- 3.3. Density estimates and a concentration lemma -- 3.4. Examples -- Chapter 4. A generalized existence result -- 4.1. Preliminary results -- 4.2. The main result -- Chapter 5. Concavity of the isoperimetric profile -- 5.1. Continuity of the isoperimetric profile -- 5.2. Approximation by smooth sets -- 5.3. Concavity of the isoperimetric profile -- Chapter 6. Sharp isoperimetric inequalities and isoperimetric rigidity -- 6.1. Convex bodies with non-degenerate asymptotic cone -- 6.2. The isoperimetric profile for small volumes -- 6.3. Isoperimetric rigidity -- Chapter 7. Asymptotic behavior of the isoperimetric profile of an unbounded convex body -- 7.1. An asymptotic isoperimetric inequality -- 7.2. Estimates on the volume growth of balls -- 7.3. Examples -- Bibliography -- Back Cover."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"--Provided by publisher.Memoirs of the American Mathematical Society Convex bodiesBoundary value problemsIsoperimetric inequalitiesCalculus of variations and optimal control; optimization -- Manifolds -- Optimization of shapes other than minimal surfacesmscConvex and discrete geometry -- General convexity -- Inequalities and extremum problemsmscConvex bodies.Boundary value problems.Isoperimetric inequalities.Calculus of variations and optimal control; optimization -- Manifolds -- Optimization of shapes other than minimal surfaces.Convex and discrete geometry -- General convexity -- Inequalities and extremum problems.516/.08516.0849Q1052A40mscLeonardi Gian Paolo1800985Ritoré Manuel0Vernadakis Efstratios1800986MiAaPQMiAaPQMiAaPQBOOK9910965974603321Isoperimetric Inequalities in Unbounded Convex Bodies4345999UNINA