A Characteristic Intersection Property of Generalized Simplices

Following R. T. Rockafellar ["Convex Analysis", Princeton University Press, Princeton (1970)], a generalized

-simplex in

{\mathbb{R}}^n R n

is defined as the direct sum of an

m m

-simplex and a simplicial

(n - m) ( n − m )

-cone,

0 \le m \le n 0 ≤ m ≤ n

. R. Fourneau ["Nonclosed simplices and quasi-simplices", Mathematika 24 (1977) 71–85] showed that a line-free

n n

-dimensional closed convex set

K \subset {{\mathbb{R}}}^n K ⊂ R n

is a generalized

n n

-simplex if and only if all

n n

-dimensional intersections

K \cap (v + K) K ∩ ( v + K )

,

v \in {{\mathbb{R}}}^n v ∈ R n

, are homothetic to

K K

. We extend this characteristic property by proving that for a pair of line-free

n n

-dimensional closed convex sets

K_1 K 1

and

K_2 K 2

in

{\mathbb{R}}^n R n

the following two conditions are equivalent: (1) all

n n

-dimensional intersections

K_1 \cap (v + K_2) K 1 ∩ ( v + K 2 )

,

v \in {{\mathbb{R}}}^n v ∈ R n

, belong to a unique homothety class of convex sets, (2)

K_1 K 1

and

K_2 K 2

are generalized

n n

-simplices whose

n n

-dimensional intersections

K_1 \cap (v + K_2) K 1 ∩ ( v + K 2 )

,

v \in {{\mathbb{R}}}^n v ∈ R n

, are homothetic to a unique generalized

n n

-simplex.