Squares and their centers

We study the relationship between the size of two sets B, S ? R², when B contains either the whole boundary or the four vertices of a square with axes-parallel sides and center in every point of S. Size refers to cardinality, Hausdorff dimension, packing dimension, or upper or lower box dimension. Perhaps surprisingly, the results vary depending on the notion of size under consideration. For example, we construct a compact set B of Hausdorff dimension 1 which contains the boundary of an axes-parallel square with center in every point in [0, 1]², prove that such a B must have packing and lower box dimension at least 7/4, and show by example that this is sharp. For more general sets of centers, the answers for packing and box counting dimensions also differ. These problems are inspired by the analogous problems for circles that were investigated by Bourgain, Marstrand and Wolff, among others.     

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.     

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.     

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.     

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.     

Weighted projective spaces and reflexive simplices

 We derive a classification algorithm for reflexive simplices in arbitrary fixed dimension. It is based on the assignment of a weight Q ? ℕ ⁿ⁺¹ to an integral n-simplex, the construction, up to an isomorphism, of all simplices with given weight Q, and the characterization in terms of the weight as to when a simplex with reduced weight is reflexive. This also yields a convex-geometric reproof of the characterization in terms of weights for weighted projective spaces to have at most Gorenstein singularities. Received: 30 March 1999 / Revised version: 18 October 2001     

Special simplices and Gorenstein toric rings

Athanasiadis [Ehrhart polynomials, simplicial polytopes, magic squares and a conjecture of Stanley, J. Reine Angew. Math., to appear.] studies an effective technique to show that Gorenstein sequences coming from compressed polytopes are unimodal. In the present paper we will use such the technique to find a rich class of Gorenstein toric rings with unimodal h-vectors arising from finite graphs.     

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.     

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.     

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).     

Triangulate flat cones on simplices

The convex hull of the vertices of a simplex and a point lying on the hyperplane spanned by the simplex is called a flat cone on the simplex. This paper proposes a natural way to simplicially triangulate flat cones on simplices and develops its applications to PL homotopy methods. This work is supported in part by the Fund of the National Committee of Education of China and in part by the National Natural Science Foundation of China.     

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.     

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.     

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.