Continuity at q = 0 of chord measures F_q(K,·)

Ascertain whether, for every convex body K in Euclidean n-space and every Borel set η contained in the unit sphere S^{n−1}, the map q ↦ F_q(K, η) is continuous at q = 0, where F_q(K,·) denotes the q-th chord measure and F_0(K,·) is defined by F_0(K,·) = ((n−1) w_{n−1})/(n w_n) S'(K,·) with S'(K,·) the (n−2)-nd area measure.

Background

The paper introduces chord measures F_q(K,·) as a new translation-invariant family of geometric measures associated with convex bodies, defined as differentials of chord-integral invariants. For q = 1, F_q reduces to the classical surface area measure S(K,·), and a limiting argument links qV_{q−1}(K,z) to mean curvature, leading to a natural definition of F_0 in terms of the area measure S'(K,·).

The authors note that establishing continuity of F_q(K,η) at q = 0 would clarify the relationship between chord measures and the classical area measures, and is relevant for understanding limiting behavior in related Minkowski-type problems.