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2012

Near equality in the two-dimensional Brunn-Minkowski inequality

Christ, Michael

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If a pair of subsets of two-dimensional Euclidean space nearly achieves equality in the Brunn-Minkowski inequality, in the sense that the measure of the associated sumset is nearly equal to the lower bound provided by the inequality, then these sets must nearly coincide with a pair of homothetic convex sets.

The proof relies on a continuum analogue of a theorem of Freiman which characterizes finite sets of integers whose sumsets are of nearly minimal size.
Small corrections and clarifications have been made in this draft.

Near equality in the two-dimensional Brunn-Minkowski inequality · Around

Built on

F. Riesz, Sur une inégalité intégrale
1930
Earlier work this paper cites.
R. Henstock and A. M. Macbeath, On the measure of sum-sets. I. The theorems of Brunn, Minkowski, and Lusternik
1953
Earlier work this paper cites.
H. Hadwiger and D. Ohmann, Brunn-Minkowskischer Satz und Isoperimetrie
1956
Earlier work this paper cites.
G. A. Freĭman, The addition of finite sets. I
1959
Earlier work this paper cites.
by same author Inverse problems of additive number theory. VI. On the addition of finite sets. III
1962
Earlier work this paper cites.

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