Etude Géométrique des Espaces Vectoriels

A geometric description of the inner aperture of a convex set

“…Our second purpose is to improve our two theorems and some results of BRONDSTED [4] by working in an arbitrary real vector space E (whole dimension can be infinite) with convex sets whose affine hull is not necessarily the whole space. Remark that the corresponding reasonings given in R" can be adapted in this genera1 case if we use the geometry of arbitrary linear spaces explained for example in [2], especially results of [1] instead of their versions in R" [5]. Therefore the proofs are left to the reader.…”

“…We shall now adopt the notations and the terminology of [2]; particularly, for a set A of a real vector space E, we shall denote by 1A, iA, aA, hA, ~A and CA the affine hull, the intrinsic core, the linear access, the algebraic hull, the margin and the asymptote cone of A, respectively [2; pp. Notice that, for a non-empty convex set A in R", •(A) coincides with P(A) when the affine hull of A is the whole space; but iflA #R", then P(A) = {0} and thus, from a geometric point of view, the study of P(A) is not very interesting in that case, in contrast to the relative notion J'(A).…”

Nonclosed simplices and quasi‐simplices

“…For the notation and terminology, we shall refer only to [1] and we shall restrict our attention to U d (d > 0), except when stated otherwise.…”

“…Let us prove (a). First of all, notice that (a) expresses the fact that S is the union of the relative interiors (internats) of the faces (" facettes " in the terminology of [1]) of S which meet S. Now, S is included in S which is the union of the relative interiors of its faces and one can delete the relative interiors of the faces P of S which do not meet S, so that Sc\J{'P:F6 &(S), 'PnS # 0}.…”

A Characterization of Homothetic Simplices

“…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]). …”