Volume and Lattice Points of Reflexive Simplices

Using new number-theoretic bounds on the denominators of unit fractions summing up to one, we show that in any dimension d ≥ 4 there is only one d-dimensional reflexive simplex having maximal volume. Moreover, only these reflexive simplices can admit an edge that has the maximal number of lattice points possible for an edge of a reflexive simplex. In general, these simplices are also expected to contain the largest number of lattice points even among all lattice polytopes with only one interior lattice point. Translated in algebro-geometric language, our main theorem yields a sharp upper bound on the anticanonical degree of d-dimensional Q-factorial Gorenstein toric Fano varieties with Picard number one, e.g., of weighted projective spaces with Gorenstein singularities.     

Covering Convex Bodies by Cylinders and Lattice Points by Flats

In connection with an unsolved problem of Bang (1951) we give a lower bound for the sum of the base volumes of cylinders covering a d-dimensional convex body in terms of the relevant basic measures of the given convex body. As an application we establish lower bounds on the number of k-dimensional flats (i.e. translates of k-dimensional linear subspaces) needed to cover all the integer points of a given convex body in d-dimensional Euclidean space for 1≤k≤d−1. K. Bezdek and A.E. Litvak are partially supported by a Natural Sciences and Engineering Research Council of Canada Discovery Grant.     

Lattice Points Inside Rational Simplices and the Casson Invariant of Brieskorn Spheres

We express the number of lattice points inside certain simplices with vertices in Q³ or Q⁴ in terms of Dedekind–Rademacher sums. This leads to an elementary proof of a formula relating the Euler characteristic of the Seiberg–Witten-Floer homology of a Brieskorn Z-homology sphere to the Casson invariant.     

Simplices by point-sliding and the Yamnitsky-Levin algorithm

Yamnitsky and Levin proposed a variant of Khachiyan's ellopsoid method for testing feasibility of systems of linear inequalities that also runs in polynomial time but uses simplices instead of ellipsoids. Starting with then-simplexS and the half-space {x¦a ^Tx }, the algorithm finds a simplexS _YL of small volume that enclosesS {x¦a ^Tx }. We interpretS _YL as a simplex obtainable by point-sliding and show that the smallest such simplex can be determined by minimizing a simple strictly convex function. We furthermore discuss some numerical results. The results suggest that the number of iterations used by our method may be considerably less than that of the standard ellipsoid method.     

Containment and Circumscribing Simplices

The following containment theorem is presented: If K and L are convex bodies such that every simplex that contains L also contains some translate of K , then in fact the body L must contain a translate of the body K . One immediate consequence of this theorem is a strengthened version of Weil's mixed-volume characterization of containment. Received October 31, 1995, and in revised form February 28, 1996.     

Inradii of Simplices

We study the following generalization of the inradius: For a convex body K in the d-dimensional Euclidean space and a linear k-plane L we define the inradius of K with respect to L by , where r(K;x+L) denotes the ordinary inradius of with respect to the affine plane x+L. We show how to determine for polytopes and use the result to estimate for the regular d-simplex T_r ^d . These estimates are optimal for all k in infinitely many dimensions and for certain k in the remaining dimensions. Received July 5, 1996, and in revised form August 8, 1996.     

Stabbing Delaunay Tetrahedralizations

A Delaunay tetrahedralization of $n$ vertices is exhibited for which a straight line can pass through the interiors of $\Theta(n^2)$ tetrahedra. This solves an open problem of Amenta, who asked whether a line can stab more than $O(n)$ tetrahedra. The construction generalizes to higher dimensions: in $d$ dimensions, a line can stab the interiors of $\Theta(n^{\lceil d / 2 \rceil})$ Delaunay $d$-simplices. The relationship between a Delaunay triangulation and a convex polytope yields another result: a two-dimensional slice of a $d$-dimensional $n$-vertex polytope can have $\Theta(n^{\lfloor d / 2 \rfloor})$ facets. This last result was first demonstrated by Amenta and Ziegler, but the construction given here is simpler and more intuitive.     

Empty Monochromatic Simplices

Let S be a k-colored (finite) set of n points in $\mathbb{R}^{d}$ , d≥3, in general position, that is, no (d+1) points of S lie in a common (d?1)-dimensional hyperplane. We count the number of empty monochromatic d-simplices determined by S, that is, simplices which have only points from one color class of S as vertices and no points of S in their interior. For 3≤k≤d we provide a lower bound of $\varOmega(n^{d-k+1+2^{-d}})$ and strengthen this to Ω(n ^d?2/3) for k=2. On the way we provide various results on triangulations of point sets in  $\mathbb{R}^{d}$ . In particular, for any constant dimension d≥3, we prove that every set of n points (n sufficiently large), in general position in $\mathbb{R}^{d}$ , admits a triangulation with at least dn+Ω(logn) simplices.     

Configurations of Flats, I: Manifolds of Points in the Projective Line

The topological manifolds arising from configurations of points in the real and complex projective lines are classified. Their topology and combinatorics are described for the real case. A general setting for the study of the spaces of configurations of flats is established and a projective duality among them is proved in its full generality.     

More Colourful Simplices

We show that any point in the convex hull of each of (d+1) sets of (d+1) points in general position in ℝ ^d is contained in at least ⌈(d+1)²/2⌉ simplices with one vertex from each set. This improves the known lower bounds for all d≥4.     

Simplices and Spectra of Graphs

In this note we show that the (n−2)-dimensional volumes of codimension 2 faces of an n-dimensional simplex are algebraically independent quantities of the volumes of its edge-lengths. The proof involves computation of the eigenvalues of Kneser graphs. We also show examples of families of simplices (of dimension 4 or greater) which show that the set of (n−2)-dimensional volumes of (n−2)-dimensional faces of a simplex do not determine its volume.     

伪对称点集与正则单形

要证明了E~n中的有序向量集是伪对称点集的充要条件.利用这一充分必要条件,得到了有关正则单形的几个等价描述,给出了伪对称点集与正则单形的关系的一个结论:设■={A_1,A_2,…,A_(n+1)}是E~n中的点集,则■是n维对称点集的充要条件是以(?)为顶点的单形是正则单形.     

Pairwise Balanced Designs Covered by Bounded Flats

We prove that for any K and d, there exists, for all sufficiently large admissible \({v}\), a pairwise balanced design PBD\({(v, K)}\) of dimension d for which all d-point-generated flats are bounded by a constant independent of v. We also tighten a prior upper bound for K = {3, 4, 5}, in which case there are no divisibility restrictions on the number of points. One consequence of this latter result is the construction of latin squares ‘covered’ by small subsquares.     

Simplices with congruent k-faces

Let S be a non-degenerate simplex in $\mathbb{R}^{2}$. We prove that S is regular if, for some k $\in$ {1,...,n-2}, all its k-dimensional faces are congruent. On the other hand, there are non-regular simplices with the property that all their (n1)-dimensional faces are congruent.     

Simplices with Equiareal Faces

If all the edges of a d -simplex T have the same length, then T is regular. However, if d geq 3 , then it is clear that the facets of T may have the same (d-1) -volume without T being regular. Here, the question of the extent to which the equality of r -volumes of the r -faces of T implies regularity of T is investigated, the case r = d-2 proving most fruitful. Received January 30, 1999. Online publication May 19, 2000.     

Inequalities for Mean Circumscribing Simplices

The aim of this note is to strengthen and generalize an inequality of Sangwine–Yager regarding means of various quantities associated with the simplices circumscribing a convex body.     

Simplices passing through a hole

We study small holes through which regular 3-, 4-, and 5-dimensional simplices can pass.