A Characteristic Intersection Property of Generalized Simplices

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 quasi-simplices", Mathematika 24 (1977) 71--85] showed that a line-free $n$-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 line-free $n$-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.