The regularity of solutions to the L _p Gauss image problem

Feng Y., Hu S., Xu L.

We will be concerned with the Lp Gaussian Minkowski problem in Gaussian probability space, which amounts to solving a class of Monge-Ampère type equations on the sphere. In this paper, the existence of symmetric (resp. asymmetric) solutions to the problem for p≤0 (resp. p≥1) will be provided.

Feng Y., Liu W., Xu L.

Existence of symmetric solutions to the Gaussian Minkowski problem was established by Huang et al. In this paper, we show the existence of non-symmetric solutions to this problem by studying the related Monge–Ampère type equation on the sphere.

Wu C., Wu D., Xiang N.

In this paper we study the $$L_p$$ Gauss image problem, which is a generalization of the $$L_p$$ Aleksandrov problem and the Gauss image problem in convex geometry. We obtain the existence result for the $$L_p$$ Gauss image problem in two cases (i) $$p>0$$ or (ii) $$p<0$$ with the given even measures.

Chen L., Wu D., Xiang N.

Li Q., Sheng W., Ye D., Yi C.

In this paper, the extended Musielak-Orlicz-Gauss image problem is studied. Such a problem aims to characterize the Musielak-Orlicz-Gauss image measure $\widetilde{C}_{G,\Psi,\lambda}(\Omega,\cdot)$ of convex body $\Omega$ in $\mathbb{R}^{n+1}$ containing the origin (but the origin is not necessary in its interior). In particular, we provide solutions to the extended Musielak-Orlicz-Gauss image problem based on the study of suitably designed parabolic flows, and by the use of approximation technique (for general measures). Our parabolic flows involve two Musielak-Orlicz functions and hence contain many well-studied curvature flows related to Minkowski type problems as special cases. Our results not only generalize many previously known solutions to the Minkowski type and Gauss image problems, but also provide solutions to those problems in many unsolved cases.

Chen H., Li Q.

This paper concerns the L p dual Minkowski problem, which amounts to solving a class of Monge-Ampère type equations on the sphere. Our main purpose is to show the existence of solutions to the problem for all p > 0 via the study of related parabolic flows. We also discuss the problem when p < 0 . The regularity and uniqueness of the solutions are also studied.

Yongsheng J., Zhengping W., Yonghong W.

We are concerned with the planar $$L_p$$ dual Minkowski problem with indices p, q. Through the compactness analysis of an associated constrained variational problem in Sobolev space, the solvability of the planar $$L_p$$ dual Minkowski problem and the related functional inequality are established, upon which the multiple solutions to the planar $$L_p$$ dual Minkowski problem are obtained. Precisely, if $$q\ge 2$$ is even, $$p<0$$ and $$q-p>16$$ , there exist at least $$[\sqrt{q-p}-2\ ]$$ convex bodies whose $$L_p$$ dual curvature measure is equal to the standard spherical measure in the plane, where $$[\sqrt{q-p}-2\ ]$$ is the integer part of $$\sqrt{q-p}-2$$ .

Du Z., Ge X.

Chen S., Huang Y., Li Q., Liu J.

In this paper, we confirm the L p -Brunn-Minkowski inequality conjecture for p close to 1. The logarithmic-Brunn-Minkowski inequality is also verified for convex bodies close to the unit ball in Hausdorff distance.

Böröczky K.J., Lutwak E., Yang D., Zhang G., Zhao Y.

Liu Y., Lu J.

In this paper the dual Orlicz–Minkowski problem, a generalization of the L p L_p dual Minkowski problem, is studied. By studying a flow involving the Gauss curvature and support function, we obtain a new existence result of solutions to this problem for smooth measures.

Li Q., Sheng W., Wang X.

In this paper we study a contracting flow of closed, convex hypersurfaces in the Euclidean space $\mathbb R^{n+1}$ with speed $f r^{\alpha} K$, where $K$ is the Gauss curvature, $r$ is the distance from the hypersurface to the origin, and $f$ is a positive and smooth function. If $\alpha \ge n+1$, we prove that the flow exists for all time and converges smoothly after normalisation to a soliton, which is a sphere centred at the origin if $f \equiv 1$. Our argument provides a parabolic proof in the smooth category for the classical Aleksandrov problem, and resolves the dual q-Minkowski problem introduced by Huang, Lutwak, Yang and Zhang (Acta Math. 216 (2016): 325-388), for the case $q 0$, we also establish the same results for even function $f$ and origin-symmetric initial condition, but for non-symmetric $f$, counterexample is given for the above smooth convergence.

Gardner R.J., Hug D., Xing S., Ye D.

For convex bodies K in $$\mathbb {R}^n$$ containing the origin in their interiors, the general dual volume and the general dual Orlicz curvature measure $$\widetilde{C}_{G, \psi }(K, \cdot )$$ were recently introduced for certain classes of functions G and $$\psi $$. We extend these concepts to more general functions G and to compact convex sets K containing the origin (but not necessarily in their interiors). Some basic properties of the general dual volume and of the dual Orlicz curvature measure are provided which are required to study a Minkowski-type problem for the dual Orlicz curvature measure. The Minkowski problem asks to characterize Borel measures $$\mu $$ on the unit sphere for which there is a convex body K in $$\mathbb {R}^n$$ containing the origin such that $$\mu $$ equals $$\widetilde{C}_{G, \psi }(K, \cdot )$$, up to a constant. A major step in the analysis concerns discrete measures $$\mu $$, for which we prove the existence of convex polytopes containing the origin in their interiors solving the Minkowski problem. Under mild conditions on G and $$\psi $$, solutions are obtained for general measures by an approximation argument. Our results generalize several previous works and provide more precise information about the solutions of the Minkowski problem when $$\mu $$ is discrete or even.

Xiong G., Xiong J., Xu L.

Sufficient conditions are given for the existence of solutions to the discrete L p Minkowski problem for p -capacity when 0 < p < 1 and 1 < p < 2 .

Böröczky K.J., Lutwak E., Yang D., Zhang G., Zhao Y.

The dual Minkowski problem for even data asks what are the necessary and sufficient conditions on a prescribed even measure on the unit sphere for it to be the q-th dual curvature measure of an origin-symmetric convex body in R n . A full solution to this is given when 1 q n . The necessary and sufficient conditions turn out to be an explicit measure concentration condition. To obtain the results, a variational approach is used, where the functional is the sum of a dual quermassintegral and an entropy integral. The proof requires two crucial estimates. The first is an estimate of the entropy integral which is obtained by using a spherical partition. The second is a sharp estimate of the dual quermassintegrals for a carefully chosen barrier convex body.