The local logarithmic Brunn-Minkowski inequality for zonoids (Q6608555)

For convex bodies \(K,L\) in \({\mathbb R}^n\) and for \(t\in[0,1]\), let \N\[\NK^{1-t}L^t:= \{z\in{\mathbb R^n}: \langle z,x\rangle\le h_K(x)^{1-t}h_L(x)^t\mbox{ for all }x\in{\mathbb R}^n\},\N\]\Nwhere \(h_{(\cdot)}\) denotes the support function. Then \(K^{1-t}L^t \subseteq (1-t)K+tL\). The logarithmic Brunn-Minkowski inequality \N\[\N\mathrm{Vol}(K^{1-t}L^t) \ge \mathrm{Vol}(K)^{1-t}\mathrm{Vol}(L)^t\N\]\Nfor \(K,L\in{\mathcal K}^n_s\) (the set of origin-symmetric convex bodies in \({\mathbb R}^n\)) was conjectured by \textit{K. Böröczky jun.} et al. [Adv. Math. 231, No. 3--4, 1974--1997 (2012; Zbl 1258.52005)] and was proved by them for \(n=2\). One result of the present paper is the truth of this inequality if \(K\) is a zonoid (a limit of finite sums of segments, here origin-symmetric), but without information on the equality case. The inequality is deduced from a `local version' (which is equivalent for general symmetric convex bodies, but not for zonoids). The latter is proved, for a zonoid and a symmetric body, by applying the Bochner method, which was employed by the author and \textit{Y. Shenfeld} [Proc. Am. Math. Soc. 147, No. 12, 5385--5402 (2019; Zbl 1457.52008)] in connection with the Aleksandrov-Fenchel inequality. For the local version, the author also characterize the equality cases.\N\NFor the entire collection see [Zbl 1527.46003].