Carla Peri INTEGRAL GEOMETRY IN MINKOWSKI PLANE 1 ...

112and OQ is the arclength of T*, then do* 0 = || — C(j3 0 )|| do 0 .Therefore we havedo 0 do + 2l I do*do = 2(L 0 +L%)L ,0 -'O -'O -'0where LQ is the length of T*.Then we have:Minkowskian Poincare formula(4) / ndr = 2(L 0 +L*)L .J{r 0 nr#(}}In the symmetric case L 0 = LQ and (4) becomes(4') I ndr = 4L 0 L,^{r 0 nr^}exactly as in Euclidean plane.4. Kinematic density for convex sets and Santalo-type formulae.From now on we restrict our attention to bounded convex sets withboundary of class C 2 .We shall say that a convex set K 0 is congruent to a convex set K : iftheir boundaries bK 0 , bK x are congruent with respect to the relation definedin the previous section.Clearly two congruent convex sets have the same area and the sameperimeter. Following Guggenheimer [3], we assume as area of a convex setits affine area.Let K be a convex set and let P{x,y) be a fixed point on bK.If C(a) is the unit tangent vector to bK at P, then we define thekinematic density for the set of convex sets congruent to K by(5) dK = dx \dy Ada.Let K 0} K be two convex sets of area S 0 , S and perimeter L 0 , L res-