We derive q-analogues of some fundamental theorems of convex geometry, including Helly's theorem, the principal kinematic formula, and Hadwiger's characterization theorem for invariant valuations.
The essential link between convex geometry and combinatorial theory is the lattice structure of the collection of polyconvex sets; that is, the collection of all finite unions of compact convex sets in ~n. This connection was highlighted by Rota in , where a valuation characterization theorem and kinematic formula were derived for the Boolean algebra of subsets of a finite set (see also ). In the present note we pursue this theme in the context of finite vector spaces.
To begin, we review a few well-known theorems of convex geometry, whose combinatorial analogues are developed in the sections following.
Helly's theorem gives a simple condition under which a finite collection of convex sets is guaranteed to have non-empty intersection . Theorem 0.1 (Helly's theorem). Let F be a finite family of compact convex sets in R n. Suppose that, for any subset G C F such that [GI ~< n + 1 (that is, every subset of cardinality at most n + 1 of F), N K#O.