Convex Solids with Planar Homothetic Sections Through Given Points

Extending results of C. A. Rogers ["Sections and projections of convex bodies", Portugal. Math. 24 (1965) 99--103], G. R. Burton ["Sections of convex bodies", J. London Math. Soc. 12 (1976) 331--336] and G. R. Burton and P. Mani ["A characterization of the ellipsoid in terms of concurrent sections, Comment. Math. Helv. 53 (1978) 485--507] to the case of unbounded convex sets, we prove that line-free 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 2-dimensional 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 2-dimensional bounded sections of convex sets are considered.