Etude Géometrique des Espaces Vectoriels II Polyèdres et Polytopes Convexes

A Characterization of Homothetic Simplices

“…Indeed, the distance between the vertices a(ε) and a (ε) continuously tends to zero when ε → 0, while the length of the edge s(ε) does not decrease. The last contradicts condition (3). Hence e ∈ N (K 1 ) and N (K 2 ) ⊂ N (K 1 ).…”

“…Bair and Fourneau [2] proved that a closed, unbounded, and line-free Choquet simplex in E d is a convex cone whose base is a k-simplex (k ≤ d). Fourneau [9]- [11] studied nonclosed unbounded Choquet simplices in E d (see also [1] and [3]). …”

Some results on quasipolyhedral convexity

“…Applying VII, Theorem 1.6 in Bair and Fourneau (1980), a non-empty convex set C is quasipolyhedral if and only if the cone of feasible directions at x 2 C; D C (x) := fy 2 R n j x + y 2 C, for some > 0g ; is polyhedral, for every x 2 C: In particular, if K is a cone, 0 n 2 K, and D K (0 n ) = fy 2 R n j y 2 K, for all 0g = K:…”

“…and the extreme points of C are isolated, hence C turns out to be quasipolyhedral (apply, for instance, VII, Theorem 1.6 in Bair and Fourneau (1980)). Nevertheless…”

“…Next we shall prove that C is a quasipolyhedral convex set in R 3 . Although VII, Theorem 1.6 in Bair and Fourneau (1980) can be again applied, for illustrative purposes, we give an alternative proof which requires the study of the faces of C: We denote by…”