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

✍ Scribed by Luis A. Caffarelli; David Jerison; Elliott H. Lieb

Publisher
Elsevier Science
Year
1996
Tongue
English
Weight
382 KB
Volume
117
Category
Article
ISSN
0001-8708

No coin nor oath required. For personal study only.

✦ Synopsis

Suppose that 0 0 and 0 1 are convex, open subsets of R N . Denote their convex combination by

The Brunn Minkowski inequality says that (vol 0 t ) 1ÂN (1&t) vol 0 1ÂN 0 +t vol 0 1ÂN 1 for 0 t 1. Moreover, if there is equality for some t other than an endpoint, then the domains 0 1 and 0 0 are translates and dilates of each other.

Borell proved an analogue of the Brunn Minkowski inequality with capacity (defined below) in place of volume. Borell's theorem [B] says article no. 0008 193 0001-8708Â96 12.00

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