Volume of the Minkowski sums of star-shaped sets

For a compact set A ⊂ R d A \subset \mathbb {R}^d and an integer alttext="k greater-than-or-equal-to 1"> k ≥<!-- ≥ <mml:mn>1 encoding="application/x-tex">k\ge 1 , let us denote by [ stretchy="false">] = { a + ⋯<!-- ⋯ <mml:mo>: , …<!-- … <mml:mo>∈<!-- ∈ <mml:mo>} ∑<!-- ∑ <mml:mi>i encoding="application/x-tex">\begin{equation*} A[k] = \left \{a_1+\cdots +a_k: a_1, \ldots , a_k\in A\right \}=\sum _{i=1}^k \end{equation*} the Minkowski sum of alttext="k"> encoding="application/x-tex">k copies encoding="application/x-tex">A . theorem Shapley, Folkmann Starr (1969) states that alttext="StartFraction Over EndFraction right-bracket"> encoding="application/x-tex">\frac {1}{k}A[k] converges to convex hull in Hausdorff distance as tends infinity. Bobkov, Madiman Wang [Concentration, functional inequalities isoperimetry, Amer. Math. Soc., Providence, RI, 2011] conjectured volume is nondecreasing or other words, terms deficit between this convergence monotone. It was proved Fradelizi, Madiman, Marsiglietti Zvavitch [C. R. Acad. Sci. Paris 354 (2016), pp. 185–189] conjecture holds true if alttext="d encoding="application/x-tex">d=1 but fails for any 12"> 12 encoding="application/x-tex">d \geq 12 In paper we show star-shaped 2"> 2 encoding="application/x-tex">d=2 3"> 3 encoding="application/x-tex">d=3 also arbitrary dimensions 4"> 4 \ge 4 under condition left-parenthesis d minus right-parenthesis 2 right-parenthesis"> stretchy="false">( −<!-- − stretchy="false">) encoding="application/x-tex">k (d-1)(d-2) addition, investigate connected sets present counterexample generalization possibly distinct alttext="double-struck encoding="application/x-tex">\mathbb 7"> 7 7