A Mixed Volumetry for the Anisotropic Logarithmic Potential

Abstract

This paper is devoted to investigating a mixed volume from the anisotropic potential with natural logarithm as a better complement to the end point case of the most recently developed mixed volumes from the anisotropic Riesz-potential. An optimal polynomial \(\log \)-inequality is not only discovered but also applicable to produce a polynomial dual for the conjectured fundamental \(\log \)-Minkowski inequality in convex geometry analysis, whence generalizing the dual \(\log \)-Minkowski inequality for mixed volume of two star bodies.