
Journal of Convex Analysis 16 (2009), No. 2, 473486 Copyright Heldermann Verlag 2009 Convex Solids with Planar Homothetic Sections Through Given Points Val Soltan Dept. of Mathematical Sciences, George Mason University, Fairfax, VA 22030, U.S.A. vsoltan@gmu.edu [Abstractpdf] \newcommand{\R}{{\mathbb{R}}} \newcommand{\Int}{\mathrm{int\,}} Extending results of C. A. Rogers ["Sections and projections of convex bodies", Portugal. Math. 24 (1965) 99103], G. R. Burton ["Sections of convex bodies", J. London Math. Soc. 12 (1976) 331336] and G. R. Burton and P. Mani ["A characterization of the ellipsoid in terms of concurrent sections, Comment. Math. Helv. 53 (1978) 485507] to the case of unbounded convex sets, we prove that linefree closed convex sets $K_1$ and $K_2$ of dimension $n$ in $\R^n$, $n \ge 4$, are homothetic provided there are points $p_1 \in \Int K_1$ and $p_2 \in \Int K_2$ such that for every pair of parallel 2dimensional planes $L_1$ and $L_2$ through $p_1$ and $p_2$, respectively, the sections $K_1 \cap L_1$ and $K_2 \cap L_2$ are homothetic. Furthermore, if there is a homothety $f : \R^n \to \R^n$ such that $f(K_1) = K_2$ and $f(p_1) \ne p_2$, then $K_1$ and $K_2$ are convex cones or their boundaries are convex quadric surfaces. Related results on elliptic and centrally symmetric 2dimensional bounded sections of convex sets are considered. Keywords: Homothety, convex body, planar section, quadric surface. MSC: 52A20 [ Fulltextpdf (155 KB)] for subscribers only. 