Search

In this paper, we prove a variational formula of torsional rigidity under the Orlicz sum, and then introduce the Orlicz mixed torsional rigidity which yields the Orlicz-Brunn-Minkowski inequality and Orlicz-Minkowski inequality for torsional rigidity. As an application, we prove the existence of solution to the Orlicz-Minkowski problem for discrete measure and general measure associated with torsional rigidity.

In this paper, we propose and study the polar Orlicz-Minkowski problems: under what conditions on a nonzero finite measure $\mu$ and a continuous function $\varphi:(0,\infty)\rightarrow(0,\infty)$, there exists a convex body $K\in\mathcal{K}_0$ such that $K$ is an optimizer of the following optimization problems: \begin{equation*} \inf/\sup \bigg\{\int_{S^{n-1}}\varphi\big( h_L \big) \,d \mu: L \in \mathcal{K}_{0} \ \text{and}\ |L^\circ|=\omega_{n}\bigg\}. \end{equation*} The solvability of the polar Orlicz-Minkowski problems is discussed under different conditions. In particular, under certain conditions on $\varphi,$ the existence of a solution is proved for a nonzero finite measure $\mu$ on $S^{n-1}$ which is not concentrated on any hemisphere of $S^{n-1}.$ Another part of this paper deals with the $p$-capacitary Orlicz-Petty bodies. In particular, the existence of the $p$-capacitary Orlicz-Petty bodies is established and the continuity of the $p$-capacitary Orlicz-Petty bodies is proved., Comment: This paper has been accepted by Indiana University Mathematics Journal

In this paper, we propose a definition of a general mixed L p affine surface area, - n ≠ p ∈ ℝ , for multiple functions. Our denfiition is different from and is "dual" to the one in [11] by Caglar and Ye. In particular, our denfiition makes it possible to establish an integral formula for the general mixed L p affine surface area of multiple functions (see Theorem 3.1 for more precise statements). Properties of the newly introduced functional are proved such as affine invariance, and related affine isoperimetric inequalities are proved. Mathematics Subject Classification (2010): 52A20, 53A15. Keywords: Affine isoperimetric inequality, affine surface area, functional inequality, L p affine surface area, L p geominimal surface area, Blaschke-Santal o inequality

In this paper, the dual Orlicz curvature measure is proposed and its basic properties are provided. A variational formula for the dual Orlicz-quermassintegral is established in order to give a geometric interpretation of the dual Orlicz curvature measure. Based on the established variational formula, a solution to the dual Orlicz–Minkowski problem regarding the dual Orlicz curvature measure is provided.

In this paper, the inclusion measures of convex bodies are studied. Some new isoperimetric-type inequalities are established by using the technique of inner parallel bodies and mixed volumes. These inequalities give the upper and lower bounds for the inclusion measures, and show the role of the form body of a convex body in these bounds.

In this paper, the optimal problem for mixed $p$-capacities is investigated. The Orlicz and $L_q$ geominimal $p$-capacities are proposed and their properties, such as invariance under orthogonal matrices, isoperimetric type inequalities and cyclic type inequalities are provided as well. Moreover, the existence of the $p$-capacitary Orlicz–Petty bodies for multiple convex bodies is established, and the Orlicz and $L_q$ mixed geominimal $p$-capacities for multiple convex bodies are introduced. The continuity of the Orlicz mixed geominimal $p$-capacities and some isoperimetric type inequalities of the $L_q$ mixed geominimal $p$-capacities are proved.

This paper is dedicated to the Orlicz-Petty bodies. We first propose the homogeneous Orlicz affine and geominimal surface areas, and establish their basic properties such as homogeneity, affine invariance and affine isoperimetric inequalities. We also prove that the homogeneous geominimal surface areas are continuous, under certain conditions, on the set of convex bodies in terms of the Hausdorff distance. Our proofs rely on the existence of the Orlicz-Petty bodies and the uniform boundedness of the Orlicz-Petty bodies of a convergent sequence of convex bodies. Similar results for the nonhomogeneous Orlicz geominimal surface areas are proved as well.

For a measure [Formula: see text] on [Formula: see text] with density that has a positive degree of homogeneous and a positive degree of concavity, we study the Orlicz–Minkowski problem for such general measure [Formula: see text] and show the existence of the even Orlicz–Minkowski problem for [Formula: see text]. We also introduce the general Orlicz geominimal [Formula: see text]-measures and show the existence of the Orlicz-Petty bodies for the general measure [Formula: see text], and hence prove the continuity of the general Orlicz geominimal [Formula: see text]-measure.

We obtain a sharp convex mixed Lorentz-Sobolev inequality, which is the functional version of an L p Minkowski inequality. The new sharp convex mixed Lorentz-Sobolev inequality implies the sharp convex Lorentz-Sobolev inequality of Ludwig, Xiao and Zhang.

This paper deals with the isoperimetric deficit upper bound for the convex domain in a surface [Formula: see text] of constant curvature [Formula: see text] by the containment measure of a convex domain to contain another convex domain in integral geometry. Some reverse Bonnesen-style inequalities are obtained. In particular, two of them strengthen Zhou’s result in [Formula: see text] and Bottema’s result in the Euclidean plane [Formula: see text].