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

In: Inequalities

Author

Listed:
  • Luis A. Caffarelli

    (Courant Institute for the Mathematical Sciences
    School of Mathematics, Institute for Advanced Study)

  • David Jerison

    (Massachusetts Institute of Technology, Department of Mathematics)

  • Elliott H. Lieb

    (Princeton University, Departments of Mathematics and Physics)

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.

Suggested Citation

  • Luis A. Caffarelli & David Jerison & Elliott H. Lieb, 2002. "On the Case of Equality in the Brunn-Minkowski Inequality for Capacity," Springer Books, in: Michael Loss & Mary Beth Ruskai (ed.), Inequalities, pages 497-511, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-55925-9_40
    DOI: 10.1007/978-3-642-55925-9_40
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