In analogy with the classical Minkowski problem, necessary and sufficient conditions are given to assure that a given measure on the unit sphere is the cone-volume measure of the unit ball of a finite dimensional Banach space.

Necessary and sufficient conditions are given for the existence of solutions to the discrete Lp Minkowski problem for the critical case where 0 < p < 1.

In this paper, the dual Orlicz curvature measure is proposed and its basic properties are provided. A variational formula for the dual Orlicz-quermassintegral is established in order to give a geometric interpretation of the dual Orlicz curvature measure. Based on the established variational formula, a solution to the dual Orlicz-Minkowski problem regarding the dual Orlicz curvature measure is provided.

A new proof of the Mahler conjecture in R2 is given. In order to prove the result, we introduce a new method  the vertex removal method; i.e., for any origin-symmetric polygon P, there exists a linear image P contained in the unit disk B2, and there exist three contiguous vertices of P lying on the boundary of B2. We can show that the volume product of P decreases when we remove the middle vertex of the three vertices.

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