The Gauss Image Problem

The Brunn-Minkowski theory and the dual Brunn-Minkowski theory are two core theories in convex geometric analysis that center on the investigation of global geometric invariants and geometric measures associated with convex bodies. The two theories display an amazing conceptual duality that involves many dual concepts in both geometry and analysis such as dual spaces in functional analysis, polarity in convex geometry, and projection and intersection in geometric tomography; see Schneider [49, p. 507] for a lucid explanation. In the conceptual duality, a central role is assumed by the radial Gauss image K (defined immediately below) of a convex body K in euclidean n-space, R. The radial Gauss image is a map on the unit sphere, S , of R whose values are subsets of the unit sphere. It is known that Aleksandrov’s integral curvature on S 1 and spherical Lebesgue measure are “linked” via the radial Gauss image, and so are the classical surface area measure of Aleksandrov-Fenchel-Jessen and Federer’s .n 1/th curvature measure (see Schneider [49, theorem 4.2.3] and [27]). The importance of the radial Gauss image was made more evident in the recent work [27], in which the long-sought dual curvature measures (the dual counterparts of Federer’s curvature measures) were unveiled. In [27] new links were established between the Brunn-Minkowski theory and the dual Brunn-Minkowski theory by making critical use of the radial Gauss image. Motivated by the manner in which these new geometric measures are defined via the radial Gauss image, it becomes