Methods of support cones and simplices in convex programming and their applications to physicochemical systems

A variant of the embedding technique proposed earlier by the second author is suggested in which the sets to be embedded are support cones. Replacing the cones by simplices gives a modification with a polynomial convergence rate.     

Affine Baire functions on Choquet simplices

We construct a metrizable simplex X such that for each n ɛ ℕ there exists a bounded function f on ext X of Baire class n that cannot be extended to a strongly affine function of Baire class n. We show that such an example cannot be constructed via the space of harmonic functions.     

Plane sections of simplices

It is clear that the longest line segment contained in a planar triangle is one of the sides. In this paper we consider the generalization of this statement to Euclideann-space, i.e., we are concerned with the validity of the proposition P(n): A hyperplane section of ann-simplex with maximum volume is a face of the simplex. Eggleston [1] proved that P(3) is valid and we prove here (i) that P(4) is valid, and (ii) a theorem asserting that P(n) false implies P(n + 1) false. However, P(5) is false, as Walkup [3] has shown, and so the question is settled for alln.     

n-Cubes inscribed in simplices

H. Bailey and D. DeTemple [1] considered some properties of squares inscribed in triangles. In this article we generalise their results to the n-dimensional space.     

On smoothness of carrying simplices

We consider dissipative strongly competitive systems of ordinary differential equations. It is known that for a wide class of such systems there exists an invariant attracting hypersurface , called the carrying simplex. In this note we give an amenable condition for to be a submanifold-with-corners. We also provide conditions, based on a recent work of M. Benaïm (On invariant hypersurfaces of strongly monotone maps, J. Differential Equations 136 (1997), 302-319), guaranteeing that is of class .

     

On certain random simplices in

The exact density is given for the r-content of the simplicial convex hull of r + 1 independent points in ⁿ, each having a type II β distribution. The density is given in the form of an integral of Mellin-Barnes type, which even in the most general cases can be evaluated to give a series representation for the density. Some special cases are evaluated to observe the types of series that can arise. It is also shown that the r-content is asymptotically normal for large values of n, a result analogous to a result conjectured by R. E. Miles (1971, Adv. in Appl. Probab., 3 353–382).     

Orthocentric simplices and their centers

A simplex is said to be orthocentric if its altitudes intersect in a common point, called its orthocenter. In this paper it is proved that if any two of the traditional centers of an orthocentric simplex (in any dimension) coincide, then the simplex is regular. Along the way orthocentric simplices in which all facets have the same circumradius are characterized, and the possible barycentric coordinates of the orthocenter are described precisely. In particular these barycentric coordinates are used to parametrize the shapes of orthocentric simplices. The substantial, but widespread, literature on orthocentric simplices is briefly surveyed in order to place the new results in their proper context, and some of the previously known results are given with new proofs from the present perspective.     

Strong ramsey properties of simplices

In this paper we will show that every simplexX with circumradiusϱ satisfies the following geometric partition property, which proves a conjecture from [FR90]. For every positive realδ there exists a positive realσ such that everygc-colouring of then-dimensional sphere of radiusϱ+δ withχ≤(1+σ) ⁿ results in a monochromatic copy ofX.     

Some results on the geometry of Choquet simplices

In this paper, we give some properties of Choquet simplices which lead to the characterization of the closed Choquet simplices of ⁿ.The results of this paper were presented at the Conference on Convexity and Foundation of Geometry held in Haifa (10–14 March 1975).     

A new view on some characterizations of simplices

The author wishes to thank Professor R. Schneider for helpful discussion, especially for referring his attention to the useful notionincluded double cone.     

Monochromatic simplices of any volume

We give a very short proof of the following result of Graham from 1980: For any finite coloring of R^d, d≥2, and for any α>0, there is a monochromatic (d+1)-tuple that spans a simplex of volume α. Our proof also yields new estimates on the number A=A(r) defined as the minimum positive value A such that, in any r-coloring of the grid points Z² of the plane, there is a monochromatic triangle of area exactly A.     

Regular simplices passing through holes

What is the smallest circular or square wall hole that a regular tetrahedron can pass? This problem was solved by Itoh et al. (Rend Circ Mat Palermo 2(77):349–354, 2006). Then, we settled the case of equilateral triangular hole in Bárány et al. (Tetrahedra passing through a triangular hole, 2009). Motivated by these results, we consider the corresponding problems in higher dimensions. Among other results, we determine the minimum (n ? 1)-dimensional ball hole that a unit regular n-simplex can pass. The diameter of the minimum hole goes to \({3\sqrt{2}/4}\) as n tends to infinity.     

Random projections of regular simplices

Precise asymptotic formulae are obtained for the expected number ofk-faces of the orthogonal projection of a regularn-simplex inn-space onto a randomly chosen isotropic subspace of fixed dimension or codimension, as the dimensionn tends to infinity.F. Affentranger was supported by a grant from the Swiss National Foundation.     

Regular simplices and Gaussian samples

We show that if a suitable type of simplex inR ⁿ is randomly rotated and its vertices projected onto a fixed subspace, they are as a point set affine-equivalent to a Gaussian sample in that subspace. Consequently, affine-invariant statistics behave the same for both mechanisms. In particular, the facet behavior for the convex hull is the same, as observed by Affentranger and Schneider; other results of theirs are translated into new results for the convex hulls of Gaussian samples. We show conversely that the conditions on the vertices of the simplex are necessary for this equivalence. Similar results hold for randomorthogonal transformations. Yuliy Baryshnikov was supported in part by the Alexander von Humboldt-Stiftung. Richard Vitale was supported in part by ONR Grant N00014-90-J-1641 and NSF Grant DMS-9002665.     

Error diffusion on acute simplices: invariant tiles

Using the notion of compatibility between Poisson brackets and cluster structures in the coordinate rings of simple Lie groups, Gekhtman, Shapiro and Vainshtein conjectured the existence of a cluster structure for each Belavin-Drinfeld solution of the classical Yang-Baxter equation compatible with the corresponding Poisson-Lie bracket on the simple Lie group. Poisson-Lie groups are classified by the Belavin-Drinfeld classification of solutions to the classical Yang-Baxter equation. For any non-trivial Belavin-Drinfeld data of minimal size for SL _n , the companion paper constructed a cluster structure with a locally regular initial seed, which was proved to be compatible with the Poisson bracket associated with that Belavin-Drinfeld data.This paper proves the rest of the conjecture: the corresponding upper cluster algebra \(\overline {{A_\mathbb{C}}} \left( C \right)\) is naturally isomorphie to O (SL _n ), the torus determined by the BD triple generates the action of \({\left( {\mathbb{C}*} \right)^{2{k_T}}}\) on C (SL _n ), and the correspondence between Belavin-Drinfeld classes and cluster structures is one to one.     

Graph isomorphism and equality of simplices

We show that the graph isomorphism problem is equivalent to the problem of recognizing equal simplices in ? ⁿ . This result can lead to new methods in the graph isomorphism problem based on geometrical properties of simplices. In particular, relations between several well-known classes of invariants of graphs and geometrical invariants of simplices are established.     

Increasing cones,recession cones and global cones

In this paper, we discuss and compare various cones used in the economics literature to analyze arbitrage in general equilibrium models with short seling. Our main result is that under certain conditions on an economic model, the closure of the increasing cone and the closure of the global cone are both equal to the arbitrage cone, the recession cone of the preferred set. It is known that in general, the Page-Wooders increasing cone may be strictly contained in the recession cone. We demonstrate that in general – for example, under the conditions of Werner (1987) or Page and Wooders (1996a, b) – the increasing cone is larger than the global cone; in fact the closure of the global cone may be strictlycontained in the increasing cone. This shows that, except under special assumptions on the economic model, conditions based on the global cone are inadequate to ensure existence of equilibrium