On the Case of Equality in the Brunn-Minkowski Inequality for Capacity

Luis A. Caffarelli, David Jerison and Elliott H. Lieb
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Luis A. Caffarelli: Courant Institute for the Mathematical Sciences
David Jerison: Massachusetts Institute of Technology, Department of Mathematics
Elliott H. Lieb: Princeton University, Departments of Mathematics and Physics

A chapter in Inequalities, 2002, pp 497-511 from Springer

Abstract: Abstract Suppose that Ω and Ω1 are convex, open subsets of Rn. Denote their convex combination by The Brunn-Minkowski inequality says that (vol Ω)t≥ (1 -t) vol Ω0 1/N +t Vol Ω for 0≤t ≤ l. Moreover, if there is equality for some t other than an endpoint, then the domains Ω1 and Ω0 are translates and dilates of each other. Borell proved an analogue of the Brunn—Minkowski inequality with capacity (defined below) in place of volume. Borel’s theorem [B] says THEOREM A. Let Ωt= tΩ1+ (1—t)Ω0 be a convex combination of two convex subsets of RN,N≥3. Then cap The main purpose of this note is to prove.

Date: 2002
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DOI: 10.1007/978-3-642-55925-9_40

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