By using the method of mixed volumes, we give sharp bounds for inclusion measures of convex bodies in n-dimensional Euclidean space. In the special cases where the random convex body is the unit ball or when n = 3, neater and simpler bounds are obtained. All the associated inequalities proved are new isoperimetrictype inequalities.

In this article, we propose the notion of the general $p$-affine capacity and prove some basic properties for the general $p$-affine capacity, such as affine invariance and monotonicity. The newly proposed general $p$-affine capacity is compared with several classical geometric quantities, e.g., the volume, the $p$-variational capacity and the $p$-integral affine surface area. Consequently, several sharp geometric inequalities for the general $p$-affine capacity are obtained. These inequalities extend and strengthen many well-known (affine) isoperimetric and (affine) isocapacitary inequalities.

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.

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A classification of $\SLn$ contravariant, continuous function-valued valuations on convex bodies is established. Such valuations are natural extensions of $\SLn$ contravariant $L_p$ Minkowski valuations, the classification of which characterized $L_p$ projection bodies, which are fundamental in the $L_p$ Brunn-Minkowski theory, for $p \geq 1$. Hence our result will help to better understand extensions of the $L_p$ Brunn-Minkowski theory. In fact, our results characterize general projection functions which extend $L_p$ projection functions ($p$-th powers of the support functions of $L_p$ projection bodies) to projection functions in the $L_p$ Brunn-Minkowski theory for $0< p < 1$ and in the Orlicz Brunn-Minkowski theory.

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The centro-affine Minkowski problem in affine differential geometry is considered. Existence for the solution of the discrete centro-affine Minkowski problem is proved.

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The study of this paper consists of two aspects. One is characterizing the so-called circular cone convexity of f by exploiting the second-order differentiability of f L ; the other is introducing the concepts of determinant and trace associated with circular cone and establishing their basic inequalities. These results show the essential role played by the angle , which gives us a new insight when looking into properties about circular cone. MSC:26A27, 26B05, 26B35, 49J52, 90C33, 65K05.

A variational formula for the Lutwak affine surface areas j of convex bodies in Rn is established when 1 ≤ j ≤ n − 1. By using introduced new ellipsoids associated with projection functions of convex bodies, we prove a sharp isoperimetric inequality for j , which opens up a new passage to attack the longstanding Lutwak conjecture in convex geometry.

The conjecture about the Orlicz Pólya–Szegö principle posed in [42] is proved. The cases of equality are characterized in the affine Orlicz Pólya–Szegö principle with respect to Steiner symmetrization and Schwarz spherical symmetrization.

We prove two mixed versions of the Discrete Nodal Theorem of Davies et. al.[3] for bounded degree graphs, and for three-connected graphs of fixed genus g . Using this we can show that for a three-connected graph satisfying a certain volume-growth condition, the multiplicity of the g th Laplacian eigenvalue is at most $2\left [

General affine invariances related to Mahler volume are introduced. We establish their affine isoperimetric inequalities by using a symmetrization scheme that involves a total of 2n elaborately chosen Steiner symmetrizations at a time. The necessity of this scheme, as opposed to the usual Steiner symmetrization, is demonstrated with an example (see the Appendix).

A first step towards a dual Orlicz-Brunn-Minkowski theory for star sets was taken by Zhu, Zhou, and Xue \cite{ZZ, ZZX}. In this essentially independent work we provide a more general framework and results. A radial Orlicz addition of two or more star sets is proposed and a corresponding dual Orlicz-Brunn-Minkowski inequality is established. Based on a radial Orlicz linear combination of two star sets, a formula for the dual Orlicz mixed volume is derived and a corresponding dual Orlicz-Minkowski inequality proved. The inequalities proved yield as special cases the precise duals of the conjectured log-Brunn-Minkowski and log-Minkowski inequalities of B\"{o}r\"{o}czky, Lutwak, Yang, and Zhang. A new addition of star sets called radial $M$-addition is also introduced and shown to relate to the radial Orlicz addition.

The original goal of this paper is to extend the affine isoperimetric inequality and Steiner type inequality of Orlicz projection bodies (which originated to Lutwak, Yang, and Zhang [36]), from convex bodies to Lipschitz star bodies (whose radial functions are locally Lipschitz). In order to achieve it, we investigate the graph functions of the given Lipschitz star body K: Along almost all directions u, we can define the graph functions on an open dense subset of the orthogonal projection of K onto u⊥.

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

Let (V, E) denote a graph with vertex set V= V (T) and edge set E= E (). Suppose a group TL acts on V such that:(i) for all g Ti,{gu, gv} GE if and only if {u, v} E,(ii) for any two vertices and v, there is a g T such that guv. Then we say is a homogeneous graph with the associated group Ti. In other words, is vertex-transitive under the action of Ti and We can identify V with the coset space Ti/X where X={g E Ti: gvv}, for a fixed vertex v, denotes the isotropy group. We note that the Cayley graph is a special case of homogeneous graphs with X trivial. The edge set of a homogeneous graph can be described by an (edge) generating set ii H such that each edge of is of the form {v, gv} for some v V, and g . In this paper we require the generating set to be symmetric, ie, g if and only if g^ 1 K. We say that a homogeneous graph is invariant if for every vertex a K, we have aKa-1= K.

The nth power of the volume functional Vnof polytopes P in R^n, according to dimensions of the spaces spanned by any nunit outer normal vectors of P, is decomposed into nhomogeneous polynomials of degree n. A set of new sharp affine isoperimetric inequalities for these volume decomposition functionals in R^3 are established, which essentially characterize the geometric and algebraic structures of polytopes.

In this paper, we are concerned with upper bounds of eigenvalues of Laplace operator on compact Riemannian manifolds and finite graphs. While on the former the Laplace operator is generated by the Riemannian metric, on the latter it reflects combinatorial structure of a graph. Respectively, eigenvalues have many applications in geometry as well as in combinatorics and in other fields of mathematics. We develop a universal approach to upper bounds on both continuous and discrete structures based upon certain properties of the corresponding heat kernel. This approach is perhaps much more general than its realization here. Basically, we start with the following entries:(1) an underlying space M with a finite measure+;(2) a well-defined Laplace operator 2 on functions on M so that 2 is a self-adjoint operator in L2 (M,+) with a discrete spectrum;(3) if M has a boundary, then the boundary condition should be chosen so that it does not disrupt self-adjointness of 2 and is of dissipative nature;(4) a distance function dist (x, y) on M so that|{dist| 1 for an appropriate notion of gradient.

In this paper, we prove the global rigidity of sphere packings on 3-dimensional manifolds. This is a 3-dimensional analogue of the rigidity theorem of Andreev-Thurston and was conjectured by Cooper and Rivin. We also prove a global rigidity result using a combinatorial scalar curvature introduced by Ge and the author.

Computing uniformization maps for surfaces has been a challenging problem and has many practical applications. In this paper, we provide a theoretically rigorous algorithm to compute such maps via combinatorial Calabi flow for vertex scaling of polyhedral metrics on surfaces, which is an analogue of the combinatorial Yamabe flow introduced by Luo (Commun Contemp Math 6(5):765–780, 2004). To handle the singularies along the combinatorial Calabi flow, we do surgery on the flow by flipping. Using the discrete conformal theory established in Gu et al. (J Differ Geom 109(3):431–466, 2018; J Differ Geom 109(2):223–256, 2018), we prove that for any initial Euclidean or hyperbolic polyhedral metric on a closed surface, the combinatorial Calabi flow with surgery exists for all time and converges exponentially fast after finite number of surgeries. The convergence is independent of the combinatorial structure of the initial triangulation on the surface.

In this paper, we demonstrate the existence part of the discrete Orlicz Minkowski problem, which is a non-trivial extension of the discrete Lp Minkowski problem for 0 < p < 1.

Inversive distance circle packing metric was introduced by P Bowers and K Stephenson as a generalization of Thurston’s circle packing metric. They conjectured that the inversive distance circle packings are rigid. For nonnegative inversive distance, Guo proved the infinitesimal rigidity and then Luo proved the global rigidity. In this paper, based on an observation of Zhou, we prove this conjecture for inversive distance in (−1, +∞)by variational principles. We also study the global rigidity of a combinatorial curvature with respect to the inversive distance circle packing metrics where the inversive distance is in (−1, +∞).

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