Vol. 292, No. 1, 2018

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ISSN: 0030-8730
Dual mean Minkowski measures and the Grünbaum conjecture for affine diameters

Qi Guo and Gabor Toth

Vol. 292 (2018), No. 1, 117–137
Abstract

For a convex body K in a Euclidean vector space X of dimension n ( 2), we define two subarithmetic monotonic sequences {σK,k}k1 and {σK,ko}k1 of functions on the interior of K. The k-th members are “mean Minkowski measures in dimension k” which are pointwise dual: σK,ko(z) = σKz,k(z), where z intK, and Kz is the dual (polar) of K with respect to z. They are measures of (anti-)symmetry of K in the following sense:

1 σK,k(z),σK,ko(z) k + 1 2 .

The lower bound is attained if and only if K has a k-dimensional simplicial slice or simplicial projection. The upper bound is attained if and only if K is symmetric with respect to z. In 1953 Klee showed that the lower bound mK > n 1 on the Minkowski measure of K implies that there are n + 1 affine diameters meeting at a critical point z K. In 1963 Grünbaum conjectured the existence of such a point in the interior of any convex body (without any conditions). While this conjecture remains open (and difficult), as a byproduct of our study of the dual mean Minkowski measures, we show that

n mK + 1 σK,n1o(z)

always holds, and for sharp inequality Grünbaum’s conjecture is valid.

Keywords
convex body, dual, Minkowski measure, affine diameter
Mathematical Subject Classification 2010
Primary: 52A05, 52A20
Secondary: 52A41, 52B55
Milestones
Received: 14 March 2016
Revised: 9 June 2017
Accepted: 9 June 2017
Published: 22 September 2017
Authors
Qi Guo
Department of Mathematics
Suzhou University of Science and Technology
Suzhou
China
Gabor Toth
Department of Mathematics
Rutgers University
Camden, NJ
United States