Dual Orlicz–Brunn–Minkowski theory

In this paper, a dual Orlicz–Brunn–Minkowski theory is presented. An Orlicz radial sum and dual Orlicz mixed volumes are introduced. The dual Orlicz–Minkowski inequality and the dual Orlicz–Brunn–Minkowski inequality are established. The variational formula for the volume with respect to the Orlicz radial sum is proved. The equivalence between the dual Orlicz–Minkowski inequality and the dual Orlicz–Brunn–Minkowski inequality is demonstrated. Orlicz intersection bodies are defined and the Orlicz–Busemann–Petty problem is posed.     

The unit ball is an attractor of the intersection body operator

The intersection body of a ball is again a ball. So, the unit ball Bd⊂Rd is a fixed point of the intersection body operator acting on the space of all star-shaped origin symmetric bodies endowed with the Banach–Mazur distance. E. Lutwak asked if there is any other star-shaped body that satisfies this property. We show that this fixed point is a local attractor, i.e., that the iterations of the intersection body operator applied to any star-shaped origin symmetric body sufficiently close to Bd in Banach–Mazur distance converge to Bd in Banach–Mazur distance. In particular, it follows that the intersection body operator has no other fixed or periodic points in a small neighborhood of Bd. We will also discuss a harmonic analysis version of this question, which studies the Radon transforms of powers of a given function.     

The divergence theorem and the Laplacian in Minkowski space

Let (X, B) be a Minkowski space (finite-dimensional Banach space) with unit ball B. Using a Minkowski definition of unit normal to a hypersurface, a Minkowski analogue of Euclidean divergence is defined. We show that the divergence theorem holds. Using the Minkowski divergence, a Minkowski Laplacian is defined. We prove that this Laplacian is a second-order, constant-coefficient, elliptic, differential operator. Furthermore, the symbol of this Laplacian is computed and used to associate a natural Euclidean structure with (X, B).Supported, in part, by NSERC Operating Grant #4066.     

Spherical containment and the Minkowski dimension of partial orders

The recent work on circle orders generalizes to higher dimensional spheres. As spherical containment is equivalent to causal precedence in Minkowski space, we define the Minkowski dimension of a poset to be the dimension of the minimal Minkowski space into which the poset can be embedded; this isd if the poset can be represented by containment with spheresS ^d–2 and of no lower dimension. Comparing this dimension with the standard dimension of partial orders we prove that they are identical in dimension two but not in higher dimensions, while their irreducible configurations are the same in dimensions two and three. Moreover, we show that there are irreducible configurations for arbitrarily large Minkowski dimension, thus providing a lower bound for the Minkowski dimension of partial orders.     

The isoperimetrix in the dual Brunn–Minkowski theory

We introduce the dual isoperimetrix, which solves the isoperimetric problem in the dual Brunn–Minkowski theory. We then show how the dual isoperimetrix is related to the isoperimetrix from the Brunn–Minkowski theory.     

The Orlicz Brunn–Minkowski inequality

The Orlicz Brunn–Minkowski theory originated with the work of Lutwak, Yang, and Zhang in 2010. In this paper, we first introduce the Orlicz addition of convex bodies containing the origin in their interiors, and then extend the _Lp$L_p$ Brunn–Minkowski inequality to the Orlicz Brunn–Minkowski inequality. Furthermore, we extend the _Lp$L_p$ Minkowski mixed volume inequality to the Orlicz mixed volume inequality by using the Orlicz Brunn–Minkowski inequality.     

A monotonicity lemma in higher dimensions

Let K be a body of constant width in a Minkowski space (i.e., in a real finite dimensional Banach space). Then any hyperplane section S of K bounds two parts of K one of which has the same diameter as S. Furthermore, if we represent K as the union of hyperplane sections S(t), t ∈[0, 1], continuously depending on t, then the Minkowskian diameter of S(t) is a unimodal function of the variable t. These two statements (being the core of this note) can be considered as higher-dimensional extensions of the well-known monotonicity lemma from Minkowski geometry.     

A characterization of Blaschke addition

A characterization of Blaschke addition as a map between origin-symmetric convex bodies is established. This results from a new characterization of Minkowski addition as a map between origin-symmetric zonoids, combined with the use of Lévy–Prokhorov metrics. A full set of examples is provided that show the results are in a sense the best possible.     

A note on reflexivity and nonconvexity

In this short note we prove that a Banach space X is reflexive if, and only if, the Eisenfeld–Lakshmikantham measure of nonconvexity in X satisfies the Cantor property. Using this characterization, some results in best approximation and fixed point theory for reflexive Banach spaces are generalized by removing convexity requirements.     

Orlicz–John ellipsoids

The Orlicz–John ellipsoids, which are in the framework of the booming Orlicz Brunn–Minkowski theory, are introduced for the first time. It turns out that they are generalizations of the classical John ellipsoid and the evolved _Lp$L_p$ John ellipsoids. The analog of Ball's volume-ratio inequality is established for the new Orlicz–John ellipsoids. The connection between the isotropy of measures and the characterization of Orlicz–John ellipsoids is demonstrated.     

Point simpliciality in Choquet theory on nonmetrizable compact spaces

Let H be a function space on a compact space K. The set of simpliciality of H is the set of all points of K for which there exists a unique maximal representing measure. Properties of this set were studied by M. Ba?ák in the paper Point simpliciality in Choquet representation theory, Illinois J. Math. 53 (2009) 289–302, mainly for K metrizable. We study properties of the set of simpliciality for K nonmetrizable.     

Reverse inequalities for zonoids and their application

We prove inequalities for mixed volumes of zonoids with isotropic generating measures. A special case is an inequality for zonoids that is reverse to the generalized Urysohn inequality, between mean width and another intrinsic volume; here the equality case characterizes parallelepipeds. We apply this to a question from stochastic geometry. It is known that among the stationary Poisson hyperplane processes of given positive intensity in n-dimensional Euclidean space, the ones with rotation invariant distribution are characterized by the fact that they yield, for k∈{2,…,n}, the maximal intensity of the intersection process of order k. We show that, if the kth intersection density is measured in an affine-invariant way, the processes of hyperplanes with only n fixed directions are characterized by a corresponding minimum property.     

Positive definite distributions and normed spaces

We answer a question of Alex Koldobsky. We show that for each −∞<p<2 and each n?3−p there is a normed space X of dimension n which embeds in Ls if and only if −n<s?p.     

Constant Minkowskian width in terms of double normals

We extend the notion of a double normal of a convex body from smooth, strictly convex Minkowski planes to arbitrary two-dimensional real, normed, linear spaces in two different ways. Then, for both of these ways, we obtain the following characterization theorem: a convex body K in a Minkowski plane is of constant Minkowskian width iff every chord I of K splits K into two compact convex sets K₁ and K₂ such that I is a Minkowskian double normal of K₁ or K₂. Furthermore, the Euclidean version of this theorem yields a new characterization of d-dimensional Euclidean ball where d 3.     

Improved Chen–Ricci inequality for curvature-like tensors and its applications

We present Chen–Ricci inequality and improved Chen–Ricci inequality for curvature like tensors. Applying our improved Chen–Ricci inequality we study Lagrangian and Kaehlerian slant submanifolds of complex space forms, and C-totally real submanifolds of Sasakian space forms.     

Fisher information and matrix-valued valuations

All continuous and affinely contravariant matrix-valued valuations on the Sobolev space W1,2(Rn) are completely classified. It is shown that there is a unique such valuation. This valuation turns out to be the Fisher information matrix.     

An angle in Minkowski space

A new angular measure in Minkowski space is introduced. It is defined for a cone of any dimension, is additive and invariant under invertible linear transformations. For the dimension 2, it has a clear interpretation as an amount of rotation in the Minkowski space from a direction to another one.     

LG/CY correspondence: The state space isomorphism

We prove the mirror duality conjecture for the mirror pairs constructed by Berglund, Hübsch, and Krawitz. Our main tool is a cohomological LG/CY correspondence which provides a degree-preserving isomorphism between the cohomology of finite quotients of Calabi–Yau hypersurfaces inside a weighted projective space and the Fan–Jarvis–Ruan–Witten state space of the associated Landau–Ginzburg singularity theory.     

Integral-type operators on continuous function spaces on the real line

In this paper we introduce some new sequences of positive linear operators, acting on a sufficiently large space of continuous functions on the real line, which generalize Gauss–Weierstrass operators.We study their approximation properties and prove an asymptotic formula that relates such operators to a second order elliptic differential operator of the form Lu?αu^′′+βu^′+γu.Shape-preserving and regularity properties are also investigated.     

The geometry of p-convex intersection bodies

Busemann's theorem states that the intersection body of an origin-symmetric convex body is also convex. In this paper we provide a version of Busemann's theorem for p-convex bodies. We show that the intersection body of a p-convex body is q-convex for certain q. Furthermore, we discuss the sharpness of the previous result by constructing an appropriate example. This example is also used to show that IK, the intersection body of K, can be much farther away from the Euclidean ball than K. Finally, we extend these theorems to some general measure spaces with log-concave and s-concave measures.