Volume of the Minkowski sums of star-shaped sets


For a compact set A in R^d and an integer k \geq 1, let us denote by A[k] the
Minkowski sum of k copies of A. A theorem of Shapley, Folkmann and Starr
(1969) states that A[k]/k converges to the convex hull of A in Hausdorff
distance as k tends to infinity. Bobkov, Madiman and Wang (2011)
conjectured that the volume of A[k]/k is non-decreasing in k, or in
other words, in terms of the volume deficit between the convex hull of A
and A[k]/k, this convergence is monotone. It was proved by Fradelizi,
Madiman, Marsiglietti and Zvavitch (2016) that this conjecture holds
true if d=1 but fails for any d \geq 12. The main goal of this talk is to
show that the conjecture is true for any star-shaped set A in R^d for
arbitrary dimensions d \geq 1 under the condition k \geq d-1. Joint work with M.
Fradelizi and A. Zvavitch.