All SL(n) covariant vector valuations on convex polytopes in R^n are completely classified without any continuity assumptions. The moment vector turns out to be the only such valuation if n ≥ 3, while two new functionals show up in dimension two.

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In this paper, we prove that, if functions (concave) $\phi$ and (convex) $\psi$ satisfy certain conditions, the $L_{\phi}$ affine surface area is monotone increasing, while the $L_{\psi}$ affine surface area is monotone decreasing under the Steiner symmetrization. Consequently, we can prove related affine isoperimetric inequalities, under certain conditions on $\phi$ and $\psi$, without assuming that the convex body involved has centroid (or the Santal\'{o} point) at the origin.

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Recently, the duals of Federer’s curvature measures, called dual curvature measures, were discovered by Huang, Lutwak, Yang & Zhang (ACTA, 2016). In the same paper, they posed the dual Minkowski problem, the characterization problem for dual curvature measures, and proved existence results when the index, q, is in (0,n). The dual Minkowski problem includes the Aleksandrov problem (q = 0) and the logarithmic Minkowski problem (q = n) as special cases. In the current work, a complete solution to the dual Minkowski problem whenever q < 0, including both existence and uniqueness, is presented.

In this paper, Orlicz valuations compatible with $SL(n)$ transforms are classified. Unlike their $L_p$ analogs, the identity \text{operator} and the reflection operator are the only $SL(n)$ \text{compatible} Orlicz valuations (up to dilations). It turns out that the Orlicz projection body operator, the Orlicz centroid body operator and the Orlicz difference body operator are not Orlicz valuations. The property that the Orlicz difference body operator is not an Orlicz valuation plays an important role in characterizing the identity operator and the reflection operator.

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The relationship between Lp affine surface area and curvature measures is investigated. As a result, a new representation of the existing notion of Lp affine surface area depending only on curvature measures is derived. Direct proofs of the equivalence between this new representation and those previously known are provided. The proofs show that the new representation is, in a sense, “polar” to that of Lutwak’s and “dual” to that of Schutt & Werner’s.

A new proof of the Mahler conjecture in R2 is given. In order to prove the result, we introduce a new method  the vertex removal method; i.e., for any origin-symmetric polygon P, there exists a linear image P contained in the unit disk B2, and there exist three contiguous vertices of P lying on the boundary of B2. We can show that the volume product of P decreases when we remove the middle vertex of the three vertices.

TheLpAleksandrovintegralcurvatureanditscorrespondingchar- acterization problem—the Lp Aleksandrov problem—were recently introduced by Huang, Lutwak, Yang, and Zhang. The current work presents a solution to the Lp Aleksandrov problem for origin-symmetric polytopes when −1 < p < 0.

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A discrete conformality for polyhedral metrics on surfaces is introduced in this paper. It is shown that each polyhedral metric on a compact surface is discrete conformal to a constant curvature polyhedral metric which is unique up to scaling. Furthermore, the constant curvature metric can be found using a finite dimensional variational principle.

The Lp dual curvature measure was introduced by Lutwak, Yang & Zhang in an attempt to unify the Lp Brunn– Minkowski theory and the dual Brunn–Minkowski theory. The characterization problem for Lp dual curvature measure, called the Lp dual Minkowski problem, is a fundamental problem in this unifying theory. The Lp dual Minkowski problem contains the Lp Minkowski problem and the dual Minkowski problem, two major problems in modern convex geometry that remain open in general. In this paper, existence results on the Lp dual Minkowski problem in the weak sense will be provided. Moreover, existence and uniqueness of the solution in the smooth category will also be demonstrated.

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The dual Loomis-Whitney inequality provides the sharp lower bound for the volume of a convex body in terms of its $(n-1)$-dimensional coordinate sections. In this paper, some reverse forms of the dual Loomis-Whitney inequality are obtained. In particular, we show that the best universal DLW-constant for origin-symmetric planar convex bodies is $1$.

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In this paper, we consider the L p dual Minkowski problem by geometric variational method. Using anisotropic Gauss–Kronecker curvature flows, we establish the existence of smooth solutions of the L p dual Minkowski problem when pq ≥ 0 and the given data is even. If f ≡ 1, we show under some restrictions on p and q that the only even, smooth, uniformly convex solution is the unit ball.

We study the smoothness of the Siciak–Zaharjuta extremal function associated to a convex body in $\mathbb{R}^{2}$ . We also prove a formula relating the complex equilibrium measure of a convex body in $\mathbb{R}^{n}$ ($n$≥2) to that of its Robin indicatrix. The main tool we use is extremal ellipses.

For $1 \leq p < \infty$, Ludwig, Haberl and Parapatits classified $L_p$ Minkowski valuations intertwining the special linear group with additional conditions such as homogeneity and continuity. In this paper,a complete classification of $L_p$ Minkowski valuations intertwining the special linear group on polytopes without any additional conditions is established for $p \geq 1$ including $p = \infty$. For $n=3$ and $p=1$, there exist valuations not mentioned before.

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.

Recently, Huang, Lutwak, Yang & Zhang discovered the duals of Federer’s curvature measures within the dual Brunn-Minkowski theory and stated the “Minkowski problem” associated with these new measures. As they showed, this dual Minkowski problem has as special cases the Aleksandrov problem (when the index is 0) and the logarithmic Minkowski problem (when the index is the dimension of the ambient space)—two problems that were never imagined to be connected in any way. Huang, Lutwak, Yang & Zhang established sufficient conditions to guarantee existence of solution to the dual Minkowski problem in the even setting. In this work, existence of solution to the even dual Minkowski problem is established under new sufficiency conditions. It was recently shown by B ̈or ̈oczky, Henk & Pollehn that these new sufficiency conditions are also necessary.

The general volume of a star body, a notion that includes the usual volume, the $q$th dual volumes, and many previous types of dual mixed volumes, is introduced. A corresponding new general dual Orlicz curvature measure is defined that specializes to the $(p,q)$-dual curvature measures introduced recently by Lutwak, Yang, and Zhang. General variational formulas are established for the general volume of two types of Orlicz linear combinations. One of these is applied to the Minkowski problem for the new general dual Orlicz curvature measure, giving in particular a solution to the Minkowski problem posed by Lutwak, Yang, and Zhang for the $(p,q)$-dual curvature measures when $p>0$ and $q<0$. A dual Orlicz-Brunn-Minkowski inequality for general volumes is obtained, as well as dual Orlicz-Minkowski-type inequalities and uniqueness results for star bodies. Finally, a very general Minkowski-type inequality, involving two Orlicz functions, two convex bodies, and a star body, is proved, that includes as special cases several others in the literature, in particular one due to Lutwak, Yang, and Zhang for the $(p,q)$-mixed volume.

Corresponding to the Legendre ellipsoid and the LYZ ellipsoid, two new sine ellipsoids are introduced in this paper. These four ellipsoids are closely related in the Pythagorean relation and duality. Several volume inequalities and the valuation properties are obtained for two new ellipsoids.