A new affine invariant for polytopes and Schneider’s projection problem

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Cone-volume measures of polytopes (2013, background): “Obviously, U(P ) ≤ V(P ) and in [19] the problem was posed that for polytopes with centroid at the origin U(P ) is bounded from below by”
Cone-volume measure of general centered convex bodies (2016, unknown): “Zhang [30] and it is defined as”
Extremum problems for the cone volume functional of convex polytopes (2010, methodology): “An effective tool to study Schneider’s projection problem is the cone volume functional U introduced by Lutwak, Yang and Zhang [25]: If P is a convex polytope in Rn which contains the origin o in its interior, then define U(P ) by U(P )n = 1 nn ∑ ui1∧···∧uin =0 hi1 · · ·hinai1 · · ·ain, (2) where u1”
Projection bodies and valuations (2002, background): “It follows from (1) that the volume of ΠK and of the polar of ΠK are affine invariants, and there are important affine isoperimetric inequalities for these quantities (see [26], [33], [20], [10], [22], and Lutwak’s survey [21]).”
Affine images of isotropic measures (2015, background): “The polytopal case of the following problem was posed in [52].”