Areas in affine spaces III the integral geometry of affine area

Crofton Measures in Projective Finsler Spaces

“…The equation (19) has, in particular, the following consequence. For x, y ∈ R n , the generating functions of B o x and B o y coincide on the subsphere s x−y , where s u := S n−1 ∩ u ⊥ ; here u ⊥ is the linear subspace through 0 orthogonal to u.…”

“…However, a wealth of them can be constructed by a nice integral-geometric approach suggested by Busemann, around 1960 (see Busemann 20,19 ). Take any positive measure µ on the space A(n, n − 1) of hyperplanes of R n which satisfies µ({H ∈ A(n, n − 1) : p ∈ H}) = 0 for each p ∈ R n (4) and 0 < µ({H ∈ A(n, n − 1) :…”

“…An effective criterion for quasi-positive measures seems to be lacking, so that the construction does not yield a complete explicit description of the smooth projective Finsler metrics. The remaining question is essentially the same as the question for the conditions which a function g in (19) has to satisfy so that the integral defines a support function. With these remarks, we leave Hilbert's fourth problem, since our main concern is the existence of Crofton measures for higher dimensional areas instead of lengths.…”

A characteristic property of the sphere

“…H. Busemann [2] investigated affine measures where this factor is not deduced from a convex symmetric body B but it is rather arbitrarily given. From the theorem of this paper follows again an answer to the question treated by H. Busemann, whether or not an arbitrary continuous m(σ) can always be derived from a symmetric star-shaped body B as surface area of its intersections with planes through its midpoint.…”

“…We consider (2) for the different a^i = 1, 2, , ri), and we put Δa { = ^Qia^, β, τ)0Q(a if β, τ). Now, multiplying by Ja if sum-ming for i form 1 to n, and then carrying out an appropriate limit process, we obtain lim where the second integral of the numerator gives the surface-measure of the intersection of B and σ and, thus, its value equals π according to our condition, provided that | τ\ < ΎJ.…”