
Journal of Convex Analysis 22 (2015), No. 1, 145159 Copyright Heldermann Verlag 2015 Convex Hypersurfaces with Hyperplanar Intersections of Their Homothetic Copies Valeriu Soltan Dept. of Mathematical Sciences, George Mason University, 4400 University Drive, Fairfax, VA 22030, U.S.A. vsoltan@gmu.edu [Abstractpdf] Extending a wellknown characteristic property of ellipsoids, we describe all convex solids $K \subset \mathbb{R}^n$, possibly unboun\ded, with the following property: for any vector $z \in \mathbb{R}^n$ and any scalar $\lambda \ne 0$ such that $K \ne z + \lambda K$, the intersection of the boundaries of $K$ and $z + \lambda K$ lies in a hyperplane. This property is related to hyperplanarity of shadowboundaries of $K$ and central symmetricity of small 2dimensional sections of $K$. Keywords: Besicovitch, body, convex, ellipse, ellipsoid, convex, quadric, section, shadowboundary, solid. MSC: 52A20 [ Fulltextpdf (151 KB)] for subscribers only. 