Intersections of dilatates of convex bodies (Q2880679)

The present paper is part of a larger project initiated by the three authors in which they propose to investigate the properties of the following two functions: \(\alpha_K(L, \rho)\) which gives the volume of the intersection of a convex body \(K\) with a dilate \(\rho L\) of another convex body \(L\), both in \(\mathbb{R}^n\); and \(\eta_K (L, \rho)\) which gives the \((n-1)\)-dimensional Hausdorff measure of the intersection of \(K\) and the boundary of \(\rho L\). Of particular interest is the concavity of the function \(\alpha_K (L, \rho)\) for which the authors prove here several very interesting results.NEWLINENEWLINEStarting with an application of the Brunn-Minkowski theory which implies that the \(\alpha_K (L, \rho)^{1/n}\) is concave, the authors show that, in dimension \(2\), under a certain condition on the position of the origin relative to the convex bodies (satisfied, in particular, if both bodies are origin-symmetric), the function \(\alpha_K (L, \rho)\) itself is concave. However, it is shown that the exponent \(1/n\) is optimal in dimension greater or equal \(3\).NEWLINENEWLINEFurther restricting \(L\) to be the unit ball \(B\) of \(\mathbb{R}^n\), the previous \(2\)-dimensional result suggests that \(\alpha_K(B, \rho)^{1/(n-1)}\) may be concave. This is further investigated and shown to hold for \(K\) a centrally symmetric slab. For a general origin-symmetric convex body \(K\) in \(\mathbb{R}^3\), the concavity of the function \(\alpha_K(B, \rho)^{1/2}\) remains a conjecture as well as the fact that the concavity properties of \(\alpha_K(B, \rho)\) changes between dimensions \(3\) and \(4\). \ Additionally, the authors establish an isoperimetric inequality for subsets of equatorial zones of the sphere \(\mathbb{S}^2\) which is further used to deduce properties of the function \(\eta_K( B, \rho)\) in \(\mathbb{R}^3\) and of the equatorial symmetral of a convex body.