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

and

K_2 K 2

of dimension

n n

in

{\mathbb{R}}^n R n

,

n \ge 4 n ≥ 4

, are homothetic provided there are points

p_1 \in {\mathrm{int\,}}K_1 p 1 ∈ i n t   K 1

and

p_2 \in {\mathrm{int\,}}K_2 p 2 ∈ i n t   K 2

such that for every pair of parallel 2-dimensional planes

L_1 L 1

and

L_2 L 2

through

p_1 p 1

and

p_2 p 2

, respectively, the sections

K_1 \cap L_1 K 1 ∩ L 1

and

K_2 \cap L_2 K 2 ∩ L 2

are homothetic. Furthermore, if there is a homothety

f: {\mathbb{R}}^n \to {{\mathbb{R}}}^n f : R n → R n

such that

f(K_1) = K_2 f ( K 1 ) = K 2

and

f(p_1) \ne p_2 f ( p 1 ) ≠ p 2

, then

K_1 K 1

and

K_2 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.