Vol. 7, No. 7, 2014

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ISSN: 1948-206X (e-only)
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Rigidity of equality cases in Steiner's perimeter inequality

Filippo Cagnetti, Maria Colombo, Guido De Philippis and Francesco Maggi

Vol. 7 (2014), No. 7, 1535–1593
Abstract

Characterization results for equality cases and for rigidity of equality cases in Steiner’s perimeter inequality are presented. (By rigidity, we mean the situation when all equality cases are vertical translations of the Steiner symmetral under consideration.) We achieve this through the introduction of a suitable measure-theoretic notion of connectedness and a fine analysis of barycenter functions for sets of finite perimeter having segments as orthogonal sections with respect to a hyperplane.

Dedicated to Nicola Fusco, for his mentorship

Keywords
symmetrization, rigidity, equality cases
Mathematical Subject Classification 2010
Primary: 49K21
Milestones
Received: 6 September 2013
Revised: 25 June 2014
Accepted: 11 August 2014
Published: 12 December 2014
Authors
Filippo Cagnetti
Department of Mathematics
University of Sussex
Pevensey 2
Brighton
BN1 9QH
United Kingdom
Maria Colombo
Scuola Normale Superiore di Pisa
Piazza dei Cavalieri 7
I-56126 Pisa
Italy
Guido De Philippis
Institute for Applied Mathematics
University of Bonn
Endenicher Allee 60
D-53115 Bonn
Germany
Francesco Maggi
Department of Mathematics
The University of Texas at Austin
2515 Speedway Stop C1200
Austin, TX 78712-1202
United States