Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Convex Analysis 22 (2015), No. 1, 145--159Copyright 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 [Abstract-pdf] Extending a well-known 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 shadow-boundaries of $K$ and central symmetricity of small 2-dimensional sections of $K$. Keywords: Besicovitch, body, convex, ellipse, ellipsoid, convex, quadric, section, shadow-boundary, solid. MSC: 52A20 [ Fulltext-pdf  (151  KB)] for subscribers only.