We apply ideas from intersection theory on toric varieties to tropical intersection theory. We introduce mixed Minkowski weights on toric varieties which interpolate between equivariant and ordinary Chow cohomology classes on complete toric varieties. These objects fit into the framework of tropical intersection theory developed by Allermann and Rau. Standard facts about intersection theory on ...

We give a direct proof of the existence of mixed fiber bodies which is not based upon the reduction to polytopes by continuity. It uses an explicit formula for the support function of a mixed fiber body instead. With the aid of this formula one can also compute the support function of a Newton polytope of a multidimensional resultant and prove a certain monotonicity property for mixed fiber bod...

If two symmetric convex bodies K and L both have nicely bounded sections, then the intersection of random rotations of K and L is also nicely bounded. For L being a subspace, this main result immediately yields the unexpected “existence vs. prevalence” phenomenon: If K has one nicely bounded section, then most sections of K are nicely bounded. The main result represents a new connection between...

In this article, we introduce a new concept of general mixed width-integral of convex bodies, and establish some of its inequalities, such as isoperimetric inequality, Aleksandrov-Fenchel inequality, and cyclic inequality. We also consider the general width-integral of order i and show its related properties and inequalities. c ©2016 All rights reserved.

Three-dimensional mathematical problems of the elasticity theory of anisotropic piecewise homogeneous bodies are discussed. A mixed type boundary contact problem is considered where on one part of the interface, rigid contact conditions are given (jumps of the displacement and the stress vectors are known), while on the remaining part screen or crack type boundary conditions are imposed. The in...

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 th...

Associated with the notion of the mixed Lp-affine surface area for multiple convex bodies for all real p (p 6= −n) which was introduced by Ye, et al. [D. Ye, B. Zhu, J. Zhou, arXiv, 2013 (2013), 38 pages], we define the concept of the mixed Lp-dual affine surface area for multiple star bodies for all real p (p 6= −n) and establish its monotonicity inequalities and cyclic inequalities. Besides, ...

The striking analogy between mixed volumes of convex bodies and mixed discriminants of positive semidefinite matrices has repeatedly been observed. It was A. D. Aleksandrov [1] who, in his second proof of the Aleksandrov–Fenchel inequalities for the mixed volumes, first introduced the mixed discriminants of positive semidefinite quadratic forms and established some of their properties, includin...

While intersection cohomology is stable under small resolutions, both ordinary and intersection cohomology are unstable under smooth deformation of singularities. For complex projective algebraic hypersurfaces with an isolated singularity, we show that the first author’s cohomology of intersection spaces is stable under smooth deformations in all degrees except possibly the middle, and in the m...