arXiv subjects
Publications and source records attributed to Niufa Fang.
In this paper, we establish a class of generalized Poincaré-type inequalities for torsional rigidity on the boundary of a convex body of class $C^{2}_{+}$ in $\rnnn$ by using the concavity of related Brunn-Minkowski inequality.
We calculate the first order variation of the Riesz $α$-energy of a log-concave function $f$ with respect to the Asplund sum. Such a variational formula induces the Riesz $α$-energy measure of log-concave function $f$, which will be denoted by $\mathfrak{R}_α(f, \cdot)$. We pose the related Riesz $α$-energy Minkowski problem aiming to find necessary and/or sufficient conditions on a pregiven Borel measure $μ$ defined on $\Rn$ so that $μ=\mathfrak{R}_α(f,\cdot)$ for some log-concave function $f$. Assuming enough smoothness, the Riesz $α$-energy Minkowski problem reduces to a new Monge-Ampère type equation involving the Riesz $α$-potential. Moreover, this new Minkowski problem can be viewed as a functional counterpart of the recent Minkowski problem for the chord measures in integral geometry posed by Lutwak, Xi, Yang and Zhang (Comm.\ Pure\ Appl.\ Math.,\ 2024). The Riesz $α$-energy Minkowski problem will be solved under certain mild conditions on $μ$.
The variation of a class of Orlicz moments with respect to the Asplund sum within the class of log-concave functions is demonstrated. Such a variational formula naturally leads to a family of dual Orlicz curvature measures for log-concave functions. They are functional analogs of dual (Orlicz) curvature measures for convex bodies. Partial existence results for the functional dual Orlicz Minkowski problem are shown.
In this paper, the $q$-th dual curvature measure is extended to convex functions and the associated Minkowski problem is posed. A special case includes the $q$-th dual curvature measure of convex bodies which defined by Huang, Lutwak, Yang and Zhang. Existence for the functional dual Minkowski problem is showed when $q\leq0$ and the uniqueness part is obtained with some assumptions.
The Busemann-Petty problem asks whether symmetric convex bodies in the Euclidean space $\mathbb{R}^n$ with smaller central hyperplane sections necessarily have smaller volume. The solution has been completed and the answer is affirmative if $n \le 4$ and negative if $n\ge 5$. 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 $\mathbb{R}^n$, if the marginal $\int_{\mathbb{R}^n\cap H}f(x)dx$ of $f$ is smaller than the marginal $\int_{\mathbb{R}^n\cap H}g(x)dx$ of $g$ for every hyperplane $H$ passing through the origin, whether the entropy ${\rm Ent}(f)$ of $f$ is bigger than the entropy ${\rm Ent}(g)$ of $g$? The Busemann-Petty problem on entropy of log-concave functions includes the Busemann-Petty problem, hence, its answer is negative when $n\geq5$. For $2\leq n\leq4$ we give a positive answer to the Busemann-Petty problem on entropy of log-concave functions.
The aim of this paper is to develop a basic framework of the $L_p$ theory for the geometry of log-concave functions, which can be viewed as a functional "lifting" of the $L_p$ Brunn-Minkowski theory for convex bodies. To fulfill this goal, by combining the $L_p$ Asplund sum of log-concave functions for all $p>1$ and the total mass, we obtain a Prékopa-Leindler type inequality and propose a definition for the first variation of the total mass in the $L_p$ setting. Based on these, we further establish an $L_p$ Minkowski type inequality related to the first variation of the total mass and derive a variational formula which motivates the definition of our $L_p$ surface area measure for log-concave functions. Consequently, the $L_p$ Minkowski problem for log-concave functions, which aims to characterize the $L_p$ surface area measure for log-concave functions, is introduced. The existence of solutions to the $L_p$ Minkowski problem for log-concave functions is obtained for $p>1$ under some mild conditions on the pre-given Borel measures.
We define the functional LYZ ellipsoid of log-concave functions. Then we give notes appended to [6].