For a positive integer s,the projection body of an s-concave function f:R^(n)→[0,+∞),a convex body in the(n+s)-dimensional Euclidean space R^(n+s),is introduced.Associated inequalities for s-concave functions,such a...For a positive integer s,the projection body of an s-concave function f:R^(n)→[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.展开更多
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 ...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.展开更多
The Busemann-Petty problem asks whether symmetric convex bodies in the Euclidean space R^(n) with smaller central hyperplane sections necessarily have smaller volumes.The solution has been completed and the answer is ...The Busemann-Petty problem asks whether symmetric convex bodies in the Euclidean space R^(n) with smaller central hyperplane sections necessarily have smaller volumes.The solution has been completed and the answer is affirmative if n≤4 and negative if n≥5.In this paper,we investigate the Busemann-Petty problem on entropy of log-concave functions:for even log-concave functions f and g with finite positive integrals in R^(n),if the marginal∫_(R^(n))∩H^(f(x)dx)of f is smaller than the marginal∫_(R^(n))∩H^(g(x)dx)of g for every hyperplane H passing through the origin,is the entropy Ent(f)of f bigger than the entropy Ent(g)of g?The BusemannPetty problem on entropy of log-concave functions includes the Busemann-Petty problem,and hence its answer is negative when n≥5.For 2≤n≤4,we give a positive answer to the Busemann-Petty problem on entropy of log-concave functions.展开更多
基金supported by the National Natural Science Foundation of China(Nos.12001291,12071318)Chern Institute of Mathematics,Nankai University。
文摘For a positive integer s,the projection body of an s-concave function f:R^(n)→[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.
基金supported by National Natural Science Foundation of China (Grant No. 11671325)
文摘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.
基金supported by National Natural Science Foundation of China(Grant No.12001291)supported by National Natural Science Foundation of China(Grant No.12071318)the Fundamental Research Funds for the Central Universities(Grant No.531118010593)。
文摘The Busemann-Petty problem asks whether symmetric convex bodies in the Euclidean space R^(n) with smaller central hyperplane sections necessarily have smaller volumes.The solution has been completed and the answer is affirmative if n≤4 and negative if n≥5.In this paper,we investigate the Busemann-Petty problem on entropy of log-concave functions:for even log-concave functions f and g with finite positive integrals in R^(n),if the marginal∫_(R^(n))∩H^(f(x)dx)of f is smaller than the marginal∫_(R^(n))∩H^(g(x)dx)of g for every hyperplane H passing through the origin,is the entropy Ent(f)of f bigger than the entropy Ent(g)of g?The BusemannPetty problem on entropy of log-concave functions includes the Busemann-Petty problem,and hence its answer is negative when n≥5.For 2≤n≤4,we give a positive answer to the Busemann-Petty problem on entropy of log-concave functions.
