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.     

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.     

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.     

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.     

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.     

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.     

Stabbing Simplices by Points and Flats

The following result was proved by Bárány in 1982: For every d≥1, there exists c _d >0 such that for every n-point set S in ℝ ^d , there is a point p∈ℝ ^d contained in at least c _d n ^d+1−O(n ^d ) of the d-dimensional simplices spanned by S. We investigate the largest possible value of c _d . It was known that c _d ≤1/(2 ^d (d+1)!) (this estimate actually holds for every point set S). We construct sets showing that c _d ≤(d+1)^−(d+1), and we conjecture that this estimate is tight. The best known lower bound, due to Wagner, is c _d ≥γ _d :=(d ²+1)/((d+1)!(d+1) ^d+1); in his method, p can be chosen as any centerpoint of S. We construct n-point sets with a centerpoint that is contained in no more than γ _d n ^d+1+O(n ^d ) simplices spanned by S, thus showing that the approach using an arbitrary centerpoint cannot be further improved.     

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.     

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.     

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.     

关于单形的几个不等式及其应用

Using theory of distance geometry and analytic method, the problem on relations about the volumes of some simplices is studied, and some new inequalities for the volumes of simplices are established. As special cases, an inequality for the volume of the pedal simplex of a simplex and other inequalities for simplices are gotten.     

Some Inequalities for Two Simplices

In this paper, we establish some inequalities for two n-dimensional simplices in the n-dimensional Euclidean space E ⁿ .     

Edge Matrix of Hyperbolic Simplices

In this paper, by using the dual problem which was solved by Feng Luo (Geom. Dedicata 64 (1997), 277–282) and a new method, we give necessary and sufficient conditions for given (n(n+1)) /2 positive real numbers to be the edge lengths of a hyperbolic n-simplex. By using determinants, we also give necessary and sufficient conditions for given (n(n+1)) /2 positive real numbers to be the edge lengths of a spherical n-simplex.Mathematics Subject Classifications (2000). 51M04, 51M05, 51M20, 51M25, 52A38, 52A37, 52B10.     

On Minkowski Sums of Simplices

We investigate the structure of the Minkowski sum of standard simplices in mathbb R^r{{mathbb R}^r}. In particular, we investigate the one-dimensional structure, the vertices, their degrees and the edges in the Minkowski sum polytope.     

涉及两个单形的一个不等式

本文建立了涉及两个n维单形及其内部任意一点的一个含参数的几何不等式,并给出了它的应用。