On a convex operator for finite sets

Let S be a finite set with m elements in a real linear space and let J_S be a set of m intervals in R. We introduce a convex operator co(S,J_S) which generalizes the familiar concepts of the convex hull, , and the affine hull, , of S. We prove that each homothet of that is contained in can be obtained using this operator. A variety of convex subsets of with interesting combinatorial properties can also be obtained. For example, this operator can assign a regular dodecagon to the 4-element set consisting of the vertices and the orthocenter of an equilateral triangle. For two types of families J_S we give two different upper bounds for the number of vertices of the polytopes produced as co(S,J_S). Our motivation comes from a recent improvement of the well-known Gauss-Lucas theorem. It turns out that a particular convex set co(S,J_S) plays a central role in this improvement.     

Cyclic polytopes, hyperplanes, and Gray codes

We consider cyclic d-polytopes P that are realizable with vertices on the moment curve $M_d:t\longrightarrow (t,t^2,\ldots,t^d)$ of order $d\geq 3$. A hyperplane H bisects a j-face of P if H meets its relative interior. For $\ell\geq 1$, we investigate the maximum number of vertices that P can have so that for some $\ell$ hyperplanes, each j-face of P is bisected by one of the hyperplanes. For $\ell > 1$, the problem translates to the existence of certain codes, or equivalently, certain paths on the cube $\{0,1\}^\ell$.     

Note dual polytopes of rational convex polytopes

LetP ^d be a rational convex polytope with dimP=d such that the origin of ^d is contained in the interiorP – P ofP. In this paper, from a viewpoint of enumeration of certain rational points inP (which originated in Ehrhart's work), a necessary and sufficient condition for the dual polytopeP ^dual ofP to be integral is presented.This research was performed while the author was staying at Massachusetts Institute of Technology during the 1988–89 academic year.     

Farthest points and cut loci on some degenerate convex surfaces

We study farthest points and cut loci on doubly covered convex polygons, and determine them explicitly on doubly covered n-dimensional simplices.     

Characterizations of cyclic polytopes

Let P denote a simplicial convex 2m-polytope with n vertices. Then the following are equivalent: (i) P is cyclic; (ii) P satisfies Gale’s Evenness Condition; (iii) Every subpolytope of P is cyclic; (iv) P has at least 2m+2 cyclic subpolytopes with n−1 vertices if n ≥ 2m+5; (v) P is neighbourly and has n universal edges. We present an additional characterization based upon an easily described point arrangement property.     

Nontrivial lower bounds for the least common multiple of some finite sequences of integers

We present here a method which allows to derive a nontrivial lower bounds for the least common multiple of some finite sequences of integers. We obtain efficient lower bounds (which in a way are optimal) for the arithmetic progressions and lower bounds less efficient (but nontrivial) for quadratic sequences whose general term has the form un=an(n+t)+b with (a,t,b)∈Z³, a?5, t?0, gcd(a,b)=1. From this, we deduce for instance the lower bound: lcm{¹²+1,²²+1,…,n²+1}?0,32n(1,442) (for all n?1). In the last part of this article, we study the integer lcm(n,n+1,…,n+k) (k∈N, n∈N^∗). We show that it has a divisor dn,k simple in its dependence on n and k, and a multiple mn,k also simple in its dependence on n. In addition, we prove that both equalities: lcm(n,n+1,…,n+k)=dn,k and lcm(n,n+1,…,n+k)=mn,k hold for an infinitely many pairs (n,k).     

A class of bounds for convex bodies in Hilbert space

We extend to infinite dimensions a class of bounds forL _p metrics of finite-dimensional convex bodies. A generalization to arbitrary increasing convex functions is done simultaneously. The main tool is the use of Gaussian measure to effect a normalization for varying dimension. At a point in the proof we also invoke a strong law of large numbers for random sets to produce a rotational averaging.Supported in part by ONR Grant N0014-90-J-1641 and NSF Grant DMS-9002665.     

Cone-volume measures of polytopes

The cone-volume measure of a polytope with centroid at the origin is proved to satisfy the subspace concentration condition. As a consequence a conjectured (a dozen years ago) fundamental sharp affine isoperimetric inequality for the U-functional is completely established – along with its equality conditions.     

Fractional perfect b-matching polytopes I: General theory

The fractional perfect b-matching polytope of an undirected graph G is the polytope of all assignments of nonnegative real numbers to the edges of G such that the sum of the numbers over all edges incident to any vertex v   is a prescribed nonnegative number _bv$b_v$. General theorems which provide conditions for nonemptiness, give a formula for the dimension, and characterize the vertices, edges and face lattices of such polytopes are obtained. Many of these results are expressed in terms of certain spanning subgraphs of G which are associated with subsets or elements of the polytope. For example, it is shown that an element u of the fractional perfect b-matching polytope of G is a vertex of the polytope if and only if each component of the graph of u either is acyclic or else contains exactly one cycle with that cycle having odd length, where the graph of u is defined to be the spanning subgraph of G whose edges are those at which u is positive.     

Inside-out polytopes

We present a common generalization of counting lattice points in rational polytopes and the enumeration of proper graph colorings, nowhere-zero flows on graphs, magic squares and graphs, antimagic squares and graphs, compositions of an integer whose parts are partially distinct, and generalized latin squares. Our method is to generalize Ehrhart's theory of lattice-point counting to a convex polytope dissected by a hyperplane arrangement. We particularly develop the applications to graph and signed-graph coloring, compositions of an integer, and antimagic labellings.     

Coxeter matroid polytopes

If Δ is a polytope in real affine space, each edge of Δ determines a reflection in the perpendicular bisector of the edge. The exchange groupW (Δ) is the group generated by these reflections, and Δ is a (Coxeter) matroid polytope if this group is finite. This simple concept of matroid polytope turns out to be an equivalent way to define Coxeter matroids. The Gelfand-Serganova Theorem and the structure of the exchange group both give us information about the matroid polytope. We then specialize this information to the case of ordinary matroids; the matroid polytope by our definition in this case turns out to be a facet of the classical matroid polytope familiar to matroid theorists. This work was supported in part by NSA grant MDA904-95-1-1056.     

The combinatorial structure of random polytopes

Choose n random points in , let P_n be their convex hull, and denote by f_i(P_n) the number of i-dimensional faces of P_n. A general method for computing the expectation of f_i(P_n), i=0,…,d−1, is presented. This generalizes classical results of Efron (in the case i=0) and Rényi and Sulanke (in the case i=d−1) to arbitrary i. For random points chosen in a smooth convex body a limit law for f_i(P_n) is proved as n→∞. For random points chosen in a polytope the expectation of f_i(P_n) is determined as n→∞. This implies an extremal property for random points chosen in a simplex.     

The smallest regular polytopes of given rank

An abstract polytope is called regular   if its automorphism group has a single orbit on flags (maximal chains). In this paper, the regular n$n$-polytopes with the smallest number of flags are found, for every rank n>1$n>1$. With a few small exceptions, the smallest regular n$n$-polytopes come from a family of ‘tight’ polytopes with 2⋅4ⁿ⁻¹$2\cdot 4^{n-1}$ flags, one for each n$n$, with Schläfli symbol {4∣4∣?∣4}$\{4\mid 4\mid ?\mid 4\}$. Also with few exceptions, these have both the smallest number of elements, and the smallest number of edges in their Hasse diagram.     

On neighborly families of convex bodies

A family of convex bodies in E^d is called neighborly if the intersection of every two of them is (d-1)-dimensional. In the present paper we prove that there is an infinite neighborly family of centrally symmetric convex bodies in E^d, d 3, such that every two of them are affinely equivalent (i.e., there is an affine transformation mapping one of them onto another), the bodies have large groups of affine automorphisms, and the volumes of the bodies are prescribed. We also prove that there is an infinite neighborly family of centrally symmetric convex bodies in E^d such that the bodies have large groups of symmetries. These two results are answers to a problem of B. Grünbaum (1963). We prove also that there exist arbitrarily large neighborly families of similar convex d-polytopes in E^d with prescribed diameters and with arbitrarily large groups of symmetries of the polytopes.     

Irreducibility of polynomials modulo p via Newton polytopes

Ostrowski established in 1919 that an absolutely irreducible integral polynomial remains absolutely irreducible modulo all sufficiently large prime numbers. We obtain a new lower bound for the size of such primes in terms of the number of integral points in the Newton polytope of the polynomial, significantly improving previous estimates for sparse polynomials.     

Minimal and reduced pairs of convex bodies

We give a new proof for the existence and uniqueness (up to translation) of plane minimal pairs of convex bodies in a given equivalence class of the Hörmander-R»dström lattice, as well as a complete characterization of plane minimal pairs using surface area measures. Moreover, we introduce the so-called reduced pairs, which are special minimal pairs. For the plane case, we characterize reduced pairs as those pairs of convex bodies whose surface area measures are mutually singular. For higher dimensions, we give two sufficient conditions for the minimality of a pair of convex polytopes, as well as a necessary and sufficient criterion for a pair of convex polytopes to be reduced. We conclude by showing that a typical pair of convex bodies, in the sense of Baire category, is reduced, and hence the unique minimal pair in its equivalence class.     

On toric h-vectors of centrally symmetric polytopes

We prove tight lower bounds for the coefficients of the toric h-vector of an arbitrary centrally symmetric polytope generalizing previous results due to R. Stanley and the author using toric varieties. Our proof here is based on the theory of combinatorial intersection cohomology for normal fans of polytopes developed by G. Barthel, J.-P. Brasselet, K. Fieseler and L. Kaup, and independently by P. Bressler and V. Lunts. This theory is also valid for nonrational polytopes when there is no standard correspondence with toric varieties. In this way we can establish our bounds for centrally symmetric polytopes even without requiring them to be rational. Received: 24 March 2004     

Lifted generalized permutahedra and composition polynomials

Generalized permutahedra are the polytopes obtained from the permutahedron by changing the edge lengths while preserving the edge directions, possibly identifying vertices along the way. We introduce a “lifting” construction for these polytopes, which turns an n  -dimensional generalized permutahedron into an (n+1)$(n+1)$-dimensional one. We prove that this construction gives rise to Stasheff ?s multiplihedron from homotopy theory, and to the more general “nestomultiplihedra”, answering two questions of Devadoss and Forcey.