The Rogers-Shephard and Zhang's projection inequalities are two reverse, affine isoperimetric inequalities relating the volumes of a convex body and its difference body and polar projection body, respectively. Following a …
The Rogers-Shephard and Zhang's projection inequalities are two reverse, affine isoperimetric inequalities relating the volumes of a convex body and its difference body and polar projection body, respectively. Following a classical work by Schneider, both inequalities have been extended to the so-called higher-order setting. In this work, we establish the higher-order analogues for these inequalities in the setting of log-concave functions. In particular, this extends the Zhang's inequality for absolutely continuous log-concave functions. We introduce an iterated sup-convolution to tackle the Rogers-Shephard inequality.
This paper examines properties and interrelations of several concepts of generalized concavity. It shows that the class of functions that are both “generalized concave” and “generalized convex” is closely related …
This paper examines properties and interrelations of several concepts of generalized concavity. It shows that the class of functions that are both “generalized concave” and “generalized convex” is closely related to the class of monotone functions of a single variable. After excluding a certain small class of exceptions, the paper shows that, for arbitrary (perhaps not differentiable) functions, concave implies pseudoconcave, pseudoconcave implies strictly quasiconcave, and strictly quasiconcave implies quasiconcave. Several results characterizing the extreme values of generalized concave functions are given.
In this paper, we present new inequalities for log-convex functions, with some applications to operator means. The significance of the obtained results is two folded; the results themselves and the …
In this paper, we present new inequalities for log-convex functions, with some applications to operator means. The significance of the obtained results is two folded; the results themselves and the way they extend many known results in the literature.
In this paper, we obtain a generalization of a double inequality on the log-convex functions, obtained by E. Neuman [8]. Also, we rediscover some inequalities of gamma and q−gamma functions. …
In this paper, we obtain a generalization of a double inequality on the log-convex functions, obtained by E. Neuman [8]. Also, we rediscover some inequalities of gamma and q−gamma functions. Finally, we present some new inequalities of Riemann zeta function. Mathematics Subject Classification: 26D07, 26D20, 33B15, 33D05
We will prove a reverse Rogers-Shephard inequality for log-concave functions. In some particular cases, the method used for general log-concave functions can be slightly improved, allowing us to prove volume …
We will prove a reverse Rogers-Shephard inequality for log-concave functions. In some particular cases, the method used for general log-concave functions can be slightly improved, allowing us to prove volume estimates for polars of $\ell_p$-diferences of convex bodies whose polar bodies under some condition on the barycenter of their polar bodies.
We will prove a reverse Rogers-Shephard inequality for log-concave functions. In some particular cases, the method used for general log-concave functions can be slightly improved, allowing us to prove volume …
We will prove a reverse Rogers-Shephard inequality for log-concave functions. In some particular cases, the method used for general log-concave functions can be slightly improved, allowing us to prove volume estimates for polars of $\ell_p$-diferences of convex bodies whose polar bodies under some condition on the barycenter of their polar bodies.
Two types of convex functions are defined in a new way as the L,t log-convex function and the L,t log-convex function of the second type,which is a general form of …
Two types of convex functions are defined in a new way as the L,t log-convex function and the L,t log-convex function of the second type,which is a general form of the log-convex function.We show that any positive continuous functions are L,t log-convex functions and obtain a series of inequalities for these two types of L,t log-convex functions,which are generalizations of famous inequalities for general L,t log-convex functions.
We prove that the exponent of the entropy of one dimensional projections of a log-concave random vector defines a 1/5-seminorm. We make two conjectures concerning reverse entropy power inequalities in …
We prove that the exponent of the entropy of one dimensional projections of a log-concave random vector defines a 1/5-seminorm. We make two conjectures concerning reverse entropy power inequalities in the log-concave setting and discuss some examples.