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The dual Loomis-Whitney inequality provides the sharp lower bound for the volume of a convex body in terms of its $(n-1)$-dimensional coordinate sections. In this paper, some reverse forms of the dual Loomis-Whitney inequality are obtained. In particular, we show that the best universal DLW-constant for origin-symmetric planar convex bodies is $1$.

Corresponding to the Legendre ellipsoid and the LYZ ellipsoid, two new sine ellipsoids are introduced in this paper. These four ellipsoids are closely related in the Pythagorean relation and duality. Several volume inequalities and the valuation properties are obtained for two new ellipsoids.

TheLpAleksandrovintegralcurvatureanditscorrespondingchar- acterization problem—the Lp Aleksandrov problem—were recently introduced by Huang, Lutwak, Yang, and Zhang. The current work presents a solution to the Lp Aleksandrov problem for origin-symmetric polytopes when −1 < p < 0.

Recently, Huang, Lutwak, Yang & Zhang discovered the duals of Federer’s curvature measures within the dual Brunn-Minkowski theory and stated the “Minkowski problem” associated with these new measures. As they showed, this dual Minkowski problem has as special cases the Aleksandrov problem (when the index is 0) and the logarithmic Minkowski problem (when the index is the dimension of the ambient space)—two problems that were never imagined to be connected in any way. Huang, Lutwak, Yang & Zhang established sufficient conditions to guarantee existence of solution to the dual Minkowski problem in the even setting. In this work, existence of solution to the even dual Minkowski problem is established under new sufficiency conditions. It was recently shown by B ̈or ̈oczky, Henk & Pollehn that these new sufficiency conditions are also necessary.