Kinematic formulas for finite vector spaces (Q1377713)

Let \(F\) be a finite field having \(q\)-elements, where \(q\) is a positive power of a prime number and let \(V\) be a vector space over \(F\) of dimension \(n\). The paper presents some so called \(q\)-analogues of classic theorems of convex geometry. First of all, there is a \(q\)-analogue of the principal kinematics formula. There are also \(q\)-analogues of its variations. Other \(q\)-analogues concern Hadwiger's characterization theorem for invariant valuations and Helly's theorem. For instance, the analogue of Helly's theorem sounds as follows. If \(E\) is a finite family of subspaces of \(V\) such that every \(n\) of those subspaces contain a common line through the origin, then all subspaces from \(E\) contain a common line through the origin.