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

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.