Parallel sections homothety bodies with minimal Mahler volume in \(\mathbb R^n\) (Q2877805)

Let \(f: [-a,a] \to \mathbb{R}\), \(a>0\), be an even, non-negative, concave function and let \(L\) be an origin symmetric, convex body such that \(L \subset \{ y \in \mathbb{R}^n \mid y \cdot u =0\}\) for some fixed vector \(u\in \mathbb{S}^{n-1}\). The authors introduce the notion of the \textit{parallel sections homothety body with homothetic section \(L\) and generating function \(f\)} as the convex body \(K=\bigcup_{x \in [-a,a]}\{ f(x)L+xu\}\).NEWLINENEWLINEIf the body \(L\) is a zonoid, the authors prove that the product of volume of \(K\) with the volume of its polar \(K^\ast\) is bounded below by \(4^n/n!\) with equality if and only if \(L\) is an \((n-1)\)-dimensional cube or octahedron and the generating function is a positive constant. The result connects with Mahler's conjecture that the above lower bound holds for any origin symmetric convex body in \(\mathbb{R}^n\).NEWLINENEWLINEIn fact, the proof of the main theorem here relies on \textit{S. Reisner}'s result [Math. Z. 192, 339--346 (1986; Zbl 0578.52005)] (see also [\textit{Y. Gordon} et al., Proc. Am. Math. Soc. 104, No. 1, 273--276 (1988; Zbl 0663.52003)]) that Mahler's conjecture holds for zonoids with equality if and only if \(K\) is the \(n\)-cube or octahedron. The other main ingredient of the present paper is the polarity transform \(f^\ast\) studied by S. Artstein-Avidan and V. Milman as it is shown that \(K^\ast=\{\bigcup_{x'\in [-1/a, 1/a]} f^\ast(x') L^\ast +x'u\}\).