For any X [subset] P the intersection of all convex subspaces containing X is said to be the convex closure of X.

By [8], the convex closure of every apartment of [G.sub.k]([PI]) coincides with [G.sub.k]([PI]); moreover, if A is an apartment in a parabolic subspace then the convex closure of A coincides with this parabolic subspace.

The subspace S contains the convex closure of f(A), i.e.

This follows from Proposition 3.1 and the fact that every polar space is the convex closure of any frame.

The convex closure of a nonempty set S is the least (with respect to [subset or equal to] convex set that contains S.

Now, [[union].sub.i.[Sigma].I] [down arrow] {[C.sub.i]} is downward closed and therefore convex, so the convex closure {[C.sub.i] : i [element of] I} is also contained in that set.

Because [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is downward closed, and therefore convex, and because taking convex closure only adds elements to a set, the set [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] satisfies the same properties, so

Hence our

convex closure [Mathematical Expression Omitted] will in general be a smaller set of worlds than the 'convex hull' of X according to Oddie.

Note that every non-degenerate polar space is the convex closure of any two of its points at distance 2 from each other.

Now (x, y) is called a special pair, the convex closure of {x, y} is the set of points on the two lines xy and yz, which is not a symp.

Note that S has diameter 2 and is the convex closure, in S, of any two points at distance 2 in S.