Stability of the reverse Blaschke-Santaló inequality for zonoids and applications
Given a centered convex body \(A\subset {\mathbb R}^n\), the volume product \(P(A)\) is defined as follows: \(P(A) :=|A|\cdot |A^*|\); here \(|\cdot|\) denotes volume and \(A^*\) is the polar body of \(A\). If \(A\) is a centered zonoid, then it is known that \(P(A)\geq P(C^n)\), where \(C^n\) is the centered affine cube. The equality here holds in the class of centered zonoids if and only if \(A=C^n\). The Mahler conjecture for centered convex bodies states that \(P(A)\geq P(C^n)\). The main purpose of this paper is to establish a strengthening of Mahler's estimate in terms of a stability result for convex zonoids, and to apply this to a problem in stochastic geometry. From the result obtained it follows that if \(P(A)\) is close to the lower bound of \(A\), then \(A\) must be close to some centered affine cube. Namely, there is a constant \(\gamma_n=\gamma_n(n)\) such that the following is true. If \(A\subset {\mathbb R}^n\) is a centered zonoid satisfying \(P(A)\leq (1+\varepsilon)P(C^n)\) for some \(\varepsilon\in [0,1]\), then \(\delta_{\text{BM}}(A,C^n)\leq 1+\gamma_n \varepsilon ^{1/n}\); \(\delta_{\text{BM}}(\cdot,\cdot)\) being the Banach-Mazur distance. This is applied in stochastic geometry as follows: We consider the zero cell \(Z_0\) of a stationary Poisson hyperplane mosaic \(\widehat X\) in \({\mathbb R}^n\). The expected number of vertices of \(Z_0\) is known to be minimal if \(\widehat X\) is a parallel mozaic, and maximal if \(\widehat X\) is isotopic. The authors show in a quantitative form that \(\widehat X\) is close to a parallel mozaic if the expected number of vertices of \(Z_0\) is almost minimal. A similar result is obtained for the upper bound.
scientific article; zbMATH DE number 5689299
scientific article