
Journal of Convex Analysis 18 (2011), No. 2, 529543 Copyright Heldermann Verlag 2011 A Characteristic Intersection Property of Generalized Simplices Valeriu Soltan Dept. of Mathematical Sciences, George Mason University, Fairfax, VA 22030, U.S.A. vsoltan@gmu.edu [Abstractpdf] \newcommand{\R}{{\mathbb{R}}} Following R. T. Rockafellar ["Convex Analysis", Princeton University Press, Princeton (1970)], a generalized $n$simplex in $\R^n$ is defined as the direct sum of an $m$simplex and a simplicial $(n  m)$cone, $0 \le m \le n$. R. Fourneau ["Nonclosed simplices and quasisimplices", Mathematika 24 (1977) 7185] showed that a linefree $n$dimensional closed convex set $K \subset \R^n$ is a generalized $n$simplex if and only if all $n$dimensional intersections $K \cap (v + K)$, $v \in \R^n$, are homothetic to $K$. We extend this characteristic property by proving that for a pair of linefree $n$dimensional closed convex sets $K_1$ and $K_2$ in $\R^n$ the following two conditions are equivalent: (1) all $n$dimensional intersections $K_1 \cap (v + K_2)$, $v \in \R^n$, belong to a unique homothety class of convex sets, (2) $K_1$ and $K_2$ are generalized $n$simplices whose $n$dimensional intersections $K_1 \cap (v + K_2)$, $v \in \R^n$, are homothetic to a unique generalized $n$simplex. Keywords: Homothety, convex body, intersection, generalized simplex. MSC: 52A20 [ Fulltextpdf (164 KB)] for subscribers only. 