Continuity of the solution to the even logarithmic Minkowski problem in the plane

This paper investigates continuity of the solution to the even logarithmic Minkowski problem in the plane. It is shown that the weak convergence of a sequence of cone-volume measures in R~2 implies the convergence of the sequence of the corresponding origin-symmetric convex bodies in the Hausdorff metric.     

The Busemann-Petty problem on entropy of log-concave functions

Science China Mathematics - The Busemann-Petty problem asks whether symmetric convex bodies in the Euclidean space ?nwith smaller central hyperplane sections necessarily have smaller volumes....     

Projection Body and Isoperimetric Inequalities for s-Concave Functions*

For a positive integer s, the projection body of an s-concave function f : Rⁿ →[0, +∞), a convex body in the (n + s)-dimensional Euclidean space R^n+s, is introduced.Associated inequalities for s-concave functions, such as, the functional isoperimetric inequality, the functional Petty projection inequality and the functional Loomis-Whitney inequality are obtained.