Star Valuations and Dual Mixed Volumes

Since its creation by Brunn and Minkowski, what has become known as the Brunn Minkowski theory has provided powerful machinery to solve a broad variety of inverse problems with stereological data. The machinery of the Brunn Minkowski theory includes mixed volumes (of Minkowski), symmetrization techniques (such as those of Steiner and Blaschke), isoperimetric inequalities (such as the Brunn Minkowski, Minkowski, and Aleksandrov Fenchel inequalities), integral transforms (such as the cosine transform), and important auxiliary bodies associated with these transforms (such as Minkowski's projection bodies). Schneider's recent book [22] on the Brunn Minkowski theory is the best available introduction to the subject. While the Brunn Minkowski theory has proven to be of enormous value in answering inverse questions regarding projections of convex bodies onto subspaces, the theory has been of little value in answering inverse questions with data regarding intersections with subspaces. However, recent advancements have been made in the development of a dual Brunn Minkowski theory [3, 4, 5, 6, 8, 9, 11, 13, 16, 23, 26, 27, 28, 29] which has been tailored specifically for dealing with such questions. In contrast to the Brunn Minkowski theory, in the dual theory convex bodies are replaced by star-shaped sets, and projections onto subspaces are replaced by intersections with subspaces. The machinery of the dual theory includes dual mixed volumes (introduced by Lutwak [15, 16]), dual isoperimetric inequalities ([16]), and important auxiliary bodies known as intersection bodies (first introduced by Busemann for special centered convex bodies in [1]; later defined for star-shaped sets by Lutwak [16]. See also [9]). A comprehensive introduction to geometric tomography, including the dual Brunn Minkowski theory, may be found in Gardner's book [17]. Unfortunately, there still remain fundamental and foundational problems with the dual Brunn Minkowski theory. One of the most beautiful and article no. 0048