A point __\( s \)__ of a closed convex subset __\( S \)__ of __\( k \)__-space is *admissible* if there is no __\( t\in S \)__ with __\( t_i\leq s_i \)__ for all __\( i=1,\dots \)__, __\( k \)__, __\( t\neq s \)__. An example is given in which the set __\( A \)__ of admissible points is not closed.

Let __\( P \)__ be the set of vectors __\( p=(p_1,\dots \)__, __\( p_k) \)__ with __\( p_i > 0 \)__ and __\( \sum_1^k p_i=1 \)__, let __\( B(p) \)__ be the set of __\( s\in S \)__ with
__\[ (p,s)=\min_{t\in S}(p,t) ,\]__
and let __\( B=\sum B(p) \)__, so that __\( B \)__ consists of exactly those points of __\( S \)__ at which there is a supporting hyperplane whose normal has positive components.

__\( B\subset A\subset \overline{B} \)__. If __\( S \)__ is determined by a finite set, there is a finite set __\( p_1,\dots \)__, __\( p_N \)__, with __\( p_j\in P \)__, such that __\( B=\sum_{j=1}^N B(p_j) \)__, so that, since __\( B(p) \)__ is closed for fixed __\( p \)__, __\( B = A = \overline{B} \)__.