Mixed Volumes Research Articles

Dual Orlicz Mixed Affine Quermassintegrals

In this paper, a class of geometric quantities of star bodies via dual Orlicz mixed volumes are presented. It is proved that these geometric quantities are affine invariant and precisely the generalizations of their classical counterparts. Further, some related inequalities with respect to these geometric quantities are established.

A flag representation of projection functions

Abstract The kth projection function υk (K, ·) of a convex body K ⊂ ℝ d , d ≥ 3, is a function on the Grassmannian G(d, k) which measures the k-dimensional volume of the projection of K onto members of G(d, k). For k = 1 and k = d − 1, simple formulas for the projection functions exist. In particular, υ d-1(K, ·) can be written as a spherical integral with respect to the surface area measure of K. Here, we generalize this result and prove two integral representations for υk (K, ·), k = 1,..., d − 1, over flag manifolds. Whereas the first representation generalizes a result of Ambartzumian (1987), but uses a flag measure which is not continuous in K, the second representation is related to a recent flag formula for mixed volumes by Hug, Rataj and Weil (2013) and depends continuously on K.

A Mixed Volumetry for the Anisotropic Logarithmic Potential

This paper is devoted to investigating a mixed volume from the anisotropic potential with natural logarithm as a better complement to the end point case of the most recently developed mixed volumes from the anisotropic Riesz-potential. An optimal polynomial \(\log \)-inequality is not only discovered but also applicable to produce a polynomial dual for the conjectured fundamental \(\log \)-Minkowski inequality in convex geometry analysis, whence generalizing the dual \(\log \)-Minkowski inequality for mixed volume of two star bodies.

Q-dual mixed volumes and L p -intersection bodies

Abstract In this paper, we introduce the new notions of L p {L_{p}} -intersection and mixed intersection bodies. Inequalities for the q-dual volume sum of L p {L_{p}} -mixed intersection bodies are established.

On the product of cocycles in a polyhedral complex

We construct an algorithm for multiplying cochains in a polyhedral complex. It depends on the choice of a linear functional on the ambient space. The cocycles form a subring in the ring of cochains, the coboundaries form an ideal in the ring of cocycles, and the quotient ring is the cohomology ring. The multiplication algorithm depends on the geometry of the cells of the complex. For simplicial complexes (the simplest geometry of cells), it reduces to the well-known Čech algorithm. Our algorithm is of geometric origin. For example, it applies in the calculation of mixed volumes of polyhedra and the construction of stable intersections of tropical varieties. In geometry it is customary to multiply cocycles with values in the exterior algebra of the ambient space. Therefore we assume that the ring of values is supercommutative.

The role of the form body in bounds for inclusion measures

In this paper, the inclusion measures of convex bodies are studied. Some new isoperimetric-type inequalities are established by using the technique of inner parallel bodies and mixed volumes. These inequalities give the upper and lower bounds for the inclusion measures, and show the role of the form body of a convex body in these bounds.

A General Method to Determine Limiting Optimal Shapes for Edge-Isoperimetric Inequalities

For a general family of graphs on $\mathbb{Z}^n$, we translate the edge-isoperimetric problem into a continuous isoperimetric problem in $\mathbb{R}^n$. We then solve the continuous isoperimetric problem using the Brunn-Minkowski inequality and Minkowski's theorem on mixed volumes. This translation allows us to conclude, under a reasonable assumption about the discrete problem, that the shapes of the optimal sets in the discrete problem approach the shape of the optimal set in the continuous problem as the size of the set grows. The solution is the zonotope defined as the Minkowski sum of the edges of the original graph. We demonstrate the efficacy of this method by revisiting some previously solved classical edge-isoperimetric problems. We then apply our method to some discrete isoperimetric problems which had not previously been solved. The complexity of those solutions suggest that it would be quite difficult to find them using discrete methods only.

Mixing and Residence Time Distribution in an Inert Gas-Shrouded Tundish

Tracer dispersion experiments were carried out in a multi-strand tundish by injecting 1 (N) NaCl solution into water. The variation of dimensionless concentration–time curves known as C-curves and mixing times with different gas flow rates were studied. The proportions of dead, mixed, and dispersed plug volumes were calculated using the ‘modified mixed model.’ The observations were explained by analyzing the behavior of the bubble plume, incoming jet velocity, and turbulent kinetic energy within the tundish.

Segre classes as integrals over polytopes

We express the Segre class of a monomial scheme – or, more generally, a scheme monomially supported on a set of divisors cutting out complete intersections – in terms of an integral computed over an associated body in Euclidean space. The formula is in the spirit of the classical Bernstein–Kouchnirenko theorem computing intersection numbers of equivariant divisors in a torus in terms of mixed volumes, but deals with the more refined intersection-theoretic invariants given by Segre classes, and holds in the less restrictive context of ‘r.c. monomial schemes’.

A generalized FKG-inequality for compositions

We prove a Fortuin–Kasteleyn–Ginibre-type inequality for the lattice of compositions of the integer n with at most r parts. As an immediate application we get a wide generalization of the classical Alexandrov–Fenchel inequality for mixed volumes and of Teissier's inequality for mixed covolumes.

ANSYS-based algorithms for a simulation of pultrusion processes

ABSTRACTPultrusion is a continuous and cost-effective process for a production of composite structural components with a constant cross-sectional area. To provide better understanding of the pultrusion processes, to support the pultrusion tooling design and process control, as well as to increase a reliability of the simulation results, two alternative finite element modeling approaches using the same continuous model with lumped material properties for the cured composite are developed, compared, and discussed. Each of them is constructed using the general-purpose finite element software ANSYS that results in considerable savings in development time and costs, and also makes available various modeling features of the finite element package. The first procedure is developed in ANSYS Mechanical environment and is based on the mixed time integration scheme and nodal control volumes method to decouple the coupled energy and species equations. The second procedure is performed using ANSYS CFX software with the cure reaction introduced as an additional variable. To demonstrate advantages and disadvantages of the developed procedures as well as to show their accuracy and reliability for the thermo-chemical simulation of pultrusion processes, two- and three-dimensional problems have been successfully analyzed.

SHAPES OF POLYHEDRA, MIXED VOLUMES AND HYPERBOLIC GEOMETRY

We generalize to higher dimensions the Bavard–Ghys construction of the hyperbolic metric on the space of polygons with fixed directions of edges. The space of convex -dimensional polyhedra with fixed directions of facet normals has a decomposition into type cones that correspond to different combinatorial types of polyhedra. This decomposition is a subfan of the secondary fan of a vector configuration and can be analyzed with the help of Gale diagrams. We construct a family of quadratic forms on each of the type cones using the theory of mixed volumes. The Alexandrov–Fenchel inequalities ensure that these forms have exactly one positive eigenvalue. This introduces a piecewise hyperbolic structure on the space of similarity classes of polyhedra with fixed directions of facet normals. We show that some of the dihedral angles on the boundary of the resulting cone-manifold are equal to .

Mixed Volumes of Hypersimplices

In this paper we consider mixed volumes of combinations of hypersimplices. These numbers, called "mixed Eulerian numbers", were first considered by A. Postnikov and were shown to satisfy many properties related to Eulerian numbers, Catalan numbers, binomial coefficients, etc. We give a general combinatorial interpretation for mixed Eulerian numbers and prove the above properties combinatorially. In particular, we show that each mixed Eulerian number enumerates a certain set of permutations in $S_n$. We also prove several new properties of mixed Eulerian numbers using our methods. Finally, we consider a type B analogue of mixed Eulerian numbers and give an analogous combinatorial interpretation for these numbers.

Characterization of simplices via the Bezout inequality for mixed volumes

We consider the following Bezout inequality for mixed volumes: $$V(K_1,\dots,K_r,\Delta[{n-r}])V_n(\Delta)^{r-1}\leq \prod_{i=1}^r V(K_i,\Delta[{n-1}])\ \text{ for }2\leq r\leq n.$$ It was shown previously that the inequality is true for any $n$-dimensional simplex $\Delta$ and any convex bodies $K_1, \dots, K_r$ in $\mathbb{R}^n$. It was conjectured that simplices are the only convex bodies for which the inequality holds for arbitrary bodies $K_1, \dots, K_r$ in $\mathbb{R}^n$. In this paper we prove that this is indeed the case if we assume that $\Delta$ is a convex polytope. Thus the Bezout inequality characterizes simplices in the class of convex $n$-polytopes. In addition, we show that if a body $\Delta$ satisfies the Bezout inequality for all bodies $K_1, \dots, K_r$ then the boundary of $\Delta$ cannot have strict points. In particular, it cannot have points with positive Gaussian curvature.

A bound for the perimeter of inner parallel bodies

We provide a sharp lower bound for the perimeter of the inner parallel sets of a convex body Ω. The bound depends only on the perimeter and inradius r of the original body and states that|∂Ωt|≥(1−tr)+n−1|∂Ω|. In particular the bound is independent of any regularity properties of ∂Ω. As a by-product of the proof we establish precise conditions for equality. The proof, which is straightforward, is based on the construction of an extremal set for a certain optimization problem and the use of basic properties of mixed volumes.

Entropy Production and Exergy Loss During Mixing of Gases

Entropy and exergy analyses of losses during mixing of flows and volumes of gas are compared. It is shown that both methods give identical analytical expressions for an ideal gas. However, exergy analysis required more cumbersome mathematical transformations than entropy analysis in order to obtain a final result. An analysis of the expressions for determining the mixing losses always gives a positive result regardless of the temperature of the mixed flows or volumes. The final analytical expressions allow mixing losses to be determined. This is very important for thermodynamic analysis of low-temperature, especially cryogenic cycles.

A qualitative analysis of sets of trajectories of mechanical systems

The evolution of geometric measures (volume, surface area) of sets of attainability of linear controlled mechanical systems with constant parameters is studied. Lyapunov's direct method, the comparison method, and theory of mixed volumes are used. Based on the general comparison theorem, estimates are obtained for the solutions of differential equations with a generalized Hukuhara derivative that describe the evolution of regions of attainability. For linear controlled systems with one degree of freedom, the maximum boundedness conditions are obtained for the area of the set of attainability. Examples of the application of the obtained results are given.

L_p-mixed intersection bodies

In this paper, we prove a dual Bergstrom type inequality for q -dual mixed volumes. Furthermore, we introduce the Lp -mixed intersection bodies for any real number p = 0 , and establish some interesting inequalities for Lp -mixed intersection bodies involving a Minkowski type inequality and three Brunn-Minkowski type inequalities. Mathematics subject classification (2010): 52A38, 52A40.

Stability in terms of two measures for set difference equations in space

The set of trajectories for discrete dynamical systems (DDS) in the space is investigated. These sets are the solutions for difference equations in a metric space (space of nonempty convex compacts with the Hausdorff metric). On the basis of the comparison principle the general theorems on stability in terms of two measures were established. Applying the Minkowskij theory of mixed volumes, for some classes of nonlinear DDS in space the finite-dimensional comparison systems were constructed. The stability in terms of two measures and Lyapunov stability of fixed points for DDS in space were studied. The examples of studies of certain dynamical systems were given to illustrate the effectiveness of obtained results.

A type of Busemann-Petty problems for general Lp-intersection bodies

Recently, the notion of general (containing symmetric and asymmetric) Lp-intersection bodies was given. In this article, by the Lp-dual mixed volumes and the general Lp-dual Blaschke bodies, we study the Lp-dual affine surface area forms of the Busemann-Petty problems for general Lp-intersection bodies. Our works belong to a new and rapidly evolving asymmetric Lp-Brunn-Minkowski theory.