Mixed Volumes Research Articles

Mixed volumes of networks with binomial steady-states

The steady-state degree of a chemical reaction network is the number of complex steady-states for generic rate constants and initial conditions. One way to bound the steady-state degree is through the mixed volume of an associated steady-state system. In this work, we show that for partitionable binomial chemical reaction systems, whose resulting steady-state systems are given by a set of binomials and a set of linear (not necessarily binomial) conservation equations, computing the mixed volume is equivalent to finding the volume of a single mixed cell that is the translate of a parallelotope. Additionally, we give a coloring condition on cycle networks to identify reaction systems with binomial steady-state ideals. We highlight both of these theorems using a class of networks referred to as species-overlapping networks and give a formula for the mixed volume of these networks.

The support of mixed area measures involving a new class of convex bodies

Mixed volumes in n-dimensional Euclidean space are functionals of n-tuples of convex bodies K,L,C1,…,Cn−2. The Alexandrov–Fenchel inequalities are fundamental inequalities between mixed volumes of convex bodies. As very special cases they cover or imply many important inequalities between basic geometric functionals. A complete characterization of the equality cases in the Alexandrov–Fenchel inequality remains a challenging open problem. Major recent progress was made by Yair Shenfeld and Ramon van Handel [13,14], in particular they resolved the problem in the cases where C1,…,Cn−2 are polytopes, zonoids or smooth bodies (under some dimensional restriction). In [6] we introduced the class of polyoids, which are defined as limits of finite Minkowski sums of polytopes having a bounded number vertices. Polyoids encompass polytopes, zonoids and triangle bodies, and they can be characterized by means of generating measures. Based on this characterization and Shenfeld and van Handel's contribution (and under a dimensional restriction), we extended their result to polyoids (or smooth bodies). Our previous result was stated in terms of the support of the mixed area measure associated with the unit ball Bn and C1,…,Cn−2. This characterization result is completed in the present work which more generally provides a geometric description of the support of the mixed area measure of an arbitrary (n−1)-tuple of polyoids (or smooth bodies). The result thus (partially) confirms a long-standing conjecture by Rolf Schneider in the case of polyoids, and hence in particular it covers the case of zonoids and triangle bodies.

Mixed volumes and the Blaschke–Lebesgue theorem

Mixed volumes and the Blaschke–Lebesgue theorem

Algorithms for computing mixed multiplicities, mixed volumes, and sectional Milnor numbers

Algorithms for computing mixed multiplicities, mixed volumes, and sectional Milnor numbers

Lp-Steiner quermassintegrals

Inspired by an Lp Steiner formula for the Lp affine surface area proved by Tatarko and Werner, we define, in analogy to the classical Steiner formula, Lp-Steiner quermassintegrals. Special cases include the classical mixed volumes, the dual mixed volumes, the Lp affine surface areas and the mixed Lp affine surface areas. We investigate the properties of the Lp-Steiner quermassintegrals in a special class of convex bodies. In particular, we show that they are rotation and reflection invariant valuations in this class of convex bodies with a certain degree of homogeneity. Such valuations seem new and have not been observed before.

Weighted Brunn-Minkowski Theory I: On weighted surface area measures

The Brunn-Minkowski theory in convex geometry concerns, among other things, the volumes, mixed volumes, and surface area measures of convex bodies. We study generalizations of these concepts to Borel measures with density in Rn – in particular, the weighted versions of mixed volumes (the so-called mixed measures) when dealing with up to three distinct convex bodies. We then formulate and analyze weighted versions of classical surface area measures, and obtain a new integral formula for the mixed measure of three bodies. As an application, we prove a Bézout-type inequality for rotational invariant log-concave measures, generalizing a result by Artstein-Avidan, Florentin and Ostrover. The results are new and interesting even for the special case of the standard Gaussian measure.

Extremizers in Soprunov and Zvavitch's Bezout inequalities for mixed volumes

In [20], I. Soprunov and A. Zvavitch have translated the Bezout inequalities (from Algebraic Geometry) into inequalities of mixed volumes satisfied by the simplex. They conjecture this set of inequalities characterizes the simplex, among all convex bodies in Rn. Together with C. Saroglou, they proved the characterization among all polytopes [17] and, for a larger set of inequalities, among all convex bodies [18]. The conjecture remains open for n≥4. In this work, we investigate necessary conditions on the structure of the boundary of a convex body K, for K to satisfy all inequalities. In particular, we obtain a new solution of the 3-dimensional case.

Mixed Eulerian Numbers and Peterson Schubert Calculus

Abstract Let $\Phi $ be a root system. Postnikov introduced and studied the mixed $\Phi $-Eulerian numbers. These numbers indicate the mixed volumes of $\Phi $-hypersimplices. As specializations of these numbers, one can obtain the usual Eulerian numbers, the Catalan numbers, and the binomial coefficients. Recent work of Berget–Spink–Tseng gave a simple computation for the mixed $\Phi $-Eulerian numbers when $\Phi $ is of type $A$. In this paper, we connect a relation between mixed $\Phi $-Eulerian numbers and Peterson Schubert calculus. By using the connection, we provide a combinatorial model for the computation of Berget–Spink–Tseng in terms of left–right diagrams that were introduced by Abe–Horiguchi–Kuwata–Zeng for the purpose of Peterson Schubert calculus. We also derive a simple computation for the mixed $\Phi $-Eulerian numbers in arbitrary Lie types from Peterson Schubert calculus.

Subspace concentration of mixed volume measures

By the expansion formula of cone-volume measures of the Minkowski combination of convex bodies, the mixed volume measures are introduced. It is shown that they are localization of the classical Minkowski mixed volumes. We prove that the mixed volume measures of zonotopes in Rn satisfy the subspace concentration property. Applications of the mixed volume measures are presented.

Curve valuations and mixed volumes in the implicitization of rational varieties

We address the description of the tropicalization of families of rational varieties under parametrizations with prescribed support, via curve valuations. We recover and extend results by Sturmfels, Tevelev and Yu for generic coefficients, considering rational parametrizations with non-trivial denominator. The advantage of our point of view is that it can be generalized to deal with non-generic parametrizations. We provide a detailed analysis of the degree of the closed image, based on combinatorial conditions on the relative positions of the supports of the polynomials defining the parametrization. We obtain a new formula and finer bounds on the degree, when the supports of the polynomials are different. We also present a new formula and bounds for the order at the origin in case the closed image is a hypersurface.

Exclusion volumes of convex bodies in high space dimensions: applications to virial coefficients and continuum percolation

Using the concepts of mixed volumes and quermassintegrals of convex geometry, we derive an exact formula for the exclusion volume v ex(K) for a general convex body K that applies in any space dimension. While our main interests concern the rotationally-averaged exclusion volume of a convex body with respect to another convex body, we also describe some results for the exclusion volumes for convex bodies with the same orientation. We show that the sphere minimizes the dimensionless exclusion volume v ex(K)/v(K) among all convex bodies, whether randomly oriented or uniformly oriented, for any d, where v(K) is the volume of K. When the bodies have the same orientation, the simplex maximizes the dimensionless exclusion volume for any d with a large-d asymptotic scaling behavior of 22d /d 3/2, which is to be contrasted with the corresponding scaling of 2 d for the sphere. We present explicit formulas for quermassintegrals W 0(K), …, W d (K) for many different nonspherical convex bodies, including cubes, parallelepipeds, regular simplices, cross-polytopes, cylinders, spherocylinders, ellipsoids as well as lower-dimensional bodies, such as hyperplates and line segments. These results are utilized to determine the rotationally-averaged exclusion volume v ex(K) for these convex-body shapes for dimensions 2 through 12. While the sphere is the shape possessing the minimal dimensionless exclusion volume, we show that, among the convex bodies considered that are sufficiently compact, the simplex possesses the maximal v ex(K)/v(K) with a scaling behavior of 21.6618…d . Subsequently, we apply these results to determine the corresponding second virial coefficient B 2(K) of the aforementioned hard hyperparticles. Our results are also applied to compute estimates of the continuum percolation threshold η c derived previously by the authors for systems of identical overlapping convex bodies. We conjecture that overlapping spheres possess the maximal value of η c among all identical nonzero-volume convex overlapping bodies for d ⩾ 2, randomly or uniformly oriented, and that, among all identical, oriented nonzero-volume convex bodies, overlapping simplices have the minimal value of η c for d ⩾ 2.

Convex bodies and graded families of monomial ideals

We show that the mixed volumes of arbitrary convex bodies are equal to mixed multiplicities of graded families of monomial ideals, and to normalized limits of mixed multiplicities of monomial ideals. This result evinces the close relation between the theories of mixed volumes from convex geometry and mixed multiplicities from commutative algebra.

Graduation-inspired synchronization for industry 4.0 planning, scheduling, and execution

Manufacturing complexity and uncertainty still perplex scholars and practitioners, especially in the modern fast-changing market with diversified customer demand, growing product varieties, and mixed production volumes. Emerging Industry 4.0 technology helps resolve the complexity and uncertainty as it dramatically enhances the connectivity, visibility, and traceability via real-time data of production environments in both physical and cyber aspects. However, the management paradigms inherited from the last industrial revolution for decision-making in Planning, Scheduling, and Execution (PSE) are not ready to tap the potentials of Industry 4.0 real-time data to break the bottleneck of complexity and uncertainty. The deepening divide between technological innovations and methodological advancements inevitably hinders progress towards Industry 4.0 manufacturing. Graduation Manufacturing System (GMS) is a novel paradigm that provides flexible and resilient principles for Industry4.0 PSE, but the decision models of GMS have not been discussed in the literature. This paper proposes Graduation-inspired Synchronization (GiSync) framework as the PSE decision mechanism under GMS. GiSync leverages the combined strengths of real-time data collected by advanced technologies and the innovative management philosophy inspired by graduation ceremonies to facilitate real-time, flexible, and resilient decision-making for Industry 4.0 PSE. It includes proximity-based order-job synchronization, suitability-based job-workstation synchronization, and Out-of-Order-based operation-resource synchronization. A case study shows that GiSync outperforms the others on average and displays minor variations in statistics regarding cost-efficiency, punctuality, and simultaneity measures, indicating that GiSync is more effective, stable, and resilient in stochastic environments.

How many roots of a system of random Laurent polynomials are real?

Abstract We say that a zero of a Laurent polynomial that lies on the unit circle with centre is real. We also say that a Laurent polynomial that is real on this circle is real. In contrast with ordinary polynomials, it is known that for random real Laurent polynomials of increasing degree the average proportion of real roots tends to rather than to . We show that this phenomenon of the asymptotically nonvanishing proportion of real roots also holds for systems of Laurent polynomials of several variables. The corresponding asymptotic formula is obtained in terms of the mixed volumes of certain convex compact sets determining the growth of the system of polynomials. Bibliography: 11 titles.

The extremals of Minkowski’s quadratic inequality

In a seminal paper “Volumen und Oberfläche” (1903), Minkowski introduced the basic notion of mixed volumes and the corresponding inequalities that lie at the heart of convex geometry. The fundamental importance of characterizing the extremals of these inequalities was already emphasized by Minkowski himself, but has to date only been resolved in special cases. In this paper, we completely settle the extremals of Minkowski’s quadratic inequality, confirming a conjecture of R. Schneider. Our proof is based on the representation of mixed volumes of arbitrary convex bodies as Dirichlet forms associated to certain highly degenerate elliptic operators. A key ingredient of the proof is a quantitative rigidity property associated to these operators.

Geometric inequalities for anti-blocking bodies

We study the class of (locally) anti-blocking bodies as well as some associated classes of convex bodies. For these bodies, we prove geometric inequalities regarding volumes and mixed volumes, including Godbersen’s conjecture, near-optimal bounds on Mahler volumes, Saint-Raymond-type inequalities on mixed volumes, and reverse Kleitman inequalities for mixed volumes. We apply our results to the combinatorics of posets and prove Sidorenko-type inequalities for linear extensions of pairs of [Formula: see text]-dimensional posets. The results rely on some elegant decompositions of differences of anti-blocking bodies, which turn out to hold for anti-blocking bodies with respect to general polyhedral cones.

Minoration via mixed volumes and Cover’s problem for general channels

We give a complete solution to an open problem of Thomas Cover in 1987 about the capacity of a relay channel in the general discrete memoryless setting without any additional assumptions. The key step in our approach is to lower bound a certain soft-max of a stochastic process by convex geometry methods, which is based on two ideas: First, the soft-max is lower bounded in terms of the supremum of another process, by approximating a convex set with a polytope with bounded number of vertices. Second, using a result of Pajor, the supremum of the process is lower bounded in terms of packing numbers by means of mixed-volume inequalities (Minkowski’s first inequality).

A fully nonlinear locally constrained anisotropic curvature flow

Given a smooth positive function F∈C∞(Sn) such that the square of its positive 1-homogeneous extension on Rn+1∖{0} is uniformly convex, the Wulff shape WF is a smooth uniformly convex body in the Euclidean space Rn+1 with F being the support function of the boundary ∂WF. In this paper, we introduce the fully nonlinear locally constrained anisotropic curvature flow ∂∂tX=(1−Ek1/kσF)νF,k=2,…,nin the Euclidean space, where Ek denotes the normalized kth anisotropic mean curvature with respect to the Wulff shape WF, σF the anisotropic support function and νF the outward anisotropic unit normal of the evolving hypersurface. We show that starting from a smooth, closed and strictly convex hypersurface in Rn+1 (n≥2), the smooth solution of the flow exists for all positive time and converges smoothly and exponentially to a scaled Wulff shape. A nice feature of this flow is that it improves a certain isoperimetric ratio. Therefore by the smooth convergence of the above flow, we provide a new proof of a class of the Alexandrov–Fenchel inequalities for anisotropic mixed volumes of smooth convex domains in the Euclidean space.

Longitudinal changes in ventricular size and function are associated with death and transplantation late after the Fontan operation

BackgroundCross-sectional studies have reported that ventricular dilation and dysfunction are associated with adverse clinical outcome in Fontan patients; however, longitudinal changes and their relationship with outcome are not known.MethodsThis was a single-center retrospective analysis of Fontan patients with at least 2 cardiovascular magnetic resonance (CMR) scans without intervening interventions. Serial measures of end-diastolic volume index (EDVI), end-systolic volume index (ESVI), ejection fraction (EF), indexed mass (massi), mass-to-volume ratio, and end-systolic wall stress (ESWS) were used to estimate within-patient change over time. Changes were compared for those with and without a composite outcome (death, heart transplant, or transplant listing) as well as between patients with left (LV) and right ventricular (RV) dominance.ResultsData from 156 patients were analyzed with a mean age at 1st CMR of 17.8 ± 9.6 years. 490 CMRs were included with median of 3 CMRs/patient (range 2–9). On regression analysis with mixed effects models, volumes and ESWS increased, while mass, mass-to-volume ratio, and EF decreased over time. With a median follow-up of 10.2 years, 14% met the composite outcome. Those with the composite outcome had a greater increase in EDVI compared to those without (4.7 vs. 0.8 ml/BSA1.3/year). Compared with LV dominance, RV dominance was associated with a greater increase in ESVI (1.4 vs. 0.5 ml/BSA1.3/year), a greater decrease in EF (− 0.61%/year vs. − 0.24%/year), and a higher rate of the composite outcome (21% vs. 8%).ConclusionsVentricles in the Fontan circulation exhibit a steady decline in performance with an increase in EDVI, ESVI, and ESWS, and decrease in EF, mass index, and mass-to-volume ratio. Those with death or need for heart transplantation have a faster increase in EDVI. Patients with rapid increase in EDVI (> 5 ml/BSA1.3/year) may be at a higher risk of adverse outcomes and may benefit from closer surveillance. RV dominance is associated with worse clinical outcomes and remodeling compared to LV dominance.

Kotrbatý’s theorem on valuations and geometric inequalities for convex bodies

Very recently J. Kotrbatý[14] has proven general inequalities for translation invariant smooth valuations formally analogous to the Hodge-Riemann bilinear relations in Kähler geometry. The goal of this note is to prove several inequalities for mixed volumes of convex bodies using Kotrbatý’s theorem as the main tool.