A Log-Minkowski Inequality For Asymmetric Convexities

The classical Brunn-Minkowski inequality and Minkowski inequality are two most important geometric inequalities in the Brunn-Minkowski theory.Those two inequalities are natural extensions of the classical isoperimetric inequality.In 2012,Boroczky-Lutwak-Yang-Zhang established the log-Minkowski inequality and log-Brunn-Minkowski inequality for origin-symmetric convex bodies in the Euclidean plane R2,and conjectured that the log-Minkowski inequality and log-Brunn-Minkowski inequality would be true for origin-symmetric convex bodies in higher dimensional space.It is believed that these conjectured inequalities would be stronger than the classical Brunn-Minkowski inequality and the clas-sical Minkowski inequality that played a key role in solving uniqueness of the log-Minkowski problem.Up to now,most of results about these two conjectured inequalities are restricted on origin-symmetric convex bodies and few results are known for general nonsymmetric convex bodies.Recently,A.Stancu investigat-ed the log-Minkowski inequality for convex bodies without the symmetric as-sumption and verified the conjectured logarithmic Minkowski inequality for some special cases.Motivated by works of Boroczky-Lutwak-Yang-Zhang and A.Stancu,we mainly investigate the log-Minkowski inequalities for nonsymmetric convex bod-ies in Rn.In chapter two,we first introduce some known log-Minkowski inequal-ities and log-Brunn-Minkowski inequalities in the plane.Then we prove some log-Minkowski inequalities and log-Minkowski-type inequalities(analogues of the conjectured log-Minkowski inequality)for general nonsymmetric convex bodies in higher dimension.These inequalities are generalizations of A.Stancu.We work on the dual log-Minkowski inequality and its analogous inequalities in chapter three.In the last chapter,we obtain an inequality for the p-affine surface area which is a generation of the p-affine isoperimetric inequality.We give an approx-imate estimate of Mahler's conjecture,that is,we obtain a lower bound of the product V(K)V(K°)of the volume of convex body K and the volume of the polar body K° of K(an affine transformation invariant).