David G. Larman

Optimization of globally convex functions

Convex functions have nice properties with respect to both minimization and maximization. Similar properties are established here for functions that are permitted to have bad local behavior but are globally convex in the sense that they behave “convexly” on triples of collinear points that are widely dispersed. The results illustrate a development that seems desirable in the interest of more realistic mathematical modeling: the “globalization” of important function properties. In connection with the maximization of globally convex functions over convex bodies in a given finite-dimensional normed space E, there is interest in estimating the maximum, for points c of bodies

Largest j-simplices in d-cubes: Some relatives of the hadamard maximum determinant problem

C \subset E

The vertices of the knapsack polytope

, of the ratio between two measures of how close c comes to being an extreme point of C. Good estimates are obtained for the cases in which E is Euclidean or has the “max” norm.

Diameters of Random Graphs.

Abstract This paper studies the computationally difficult problem of finding a largest j -dimensional simplex in a given d -dimensional cube. The case in which j = d is of special interest, for it is equivalent to the Hadamard maximum determinant problem; it has been solved for infinitely many values of d but not for d = 14. (The subcase in which j = d ≡ 3 (mod 4) subsumes the famous problem on the existence of Hadamard matrices.) The known results for the case j = d are here summarized and used, but the main focus is on fixed small values of j . When j = 1, the problem is trivial, and when j = 2 or j = 3 it is here solved completely (i.e., for all d ). Beyond that, the results are fragmentary but numerous, and they lead to several attractive conjectures. Some other problems involving simplices in cubes are mentioned, and the relationship of largest simplices to D-optimal weighing designs is discussed.

Largestj-simplices inn-polytopes

Abstract The number of vertices of a polytope associated to the Knapsack integer programming problem is shown to be small. An algorithm for finding these vertices is discussed.

The Blocking Numbers of Convex Bodies

Abstract : In addition to being of interest for its own sake, the study of random graphs provides the combinatorial foundation for investigations of the average-case behavior of various graph-theoretic algorithms. Diameters of graphs are especially relevant to algorithms based on breadth-first search. The present paper deals with the family G approximate (n,E) of all labeled graphs that have n nodes and E edges.

Determination of convex bodies by certain sets of sectional volumes

Relative to a given convex bodyC, aj-simplexS inC islargest if it has maximum volume (j-measure) among allj-simplices contained inC, andS isstable (resp.rigid) if vol(S)≥vol(S′) (resp. vol(S)>vol(S′)) for eachj-simplexS′ that is obtained fromS by moving a single vertex ofS to a new position inC. This paper contains a variety of qualitative results that are related to the problems of finding a largest, a stable, or a rigidj-simplex in a givenn-dimensional convex body or convex polytope. In particular, the computational complexity of these problems is studied both for-polytopes (presented as the convex hull of a finite set of points) and forℋ-polytopes (presented as an intersection of finitely many half-spaces).

Isoradial Bodies

Abstract. Besides determining the exact blocking numbers of cubes and balls, a conditional lower bound for the blocking numbers of convex bodies is achieved. In addition, several open problems are proposed.

A combinatorial property of points and ellipsoids

Abstract Suppose that K and L are convex bodies in R n with L⊂ int K . For each hyperplane H⊂ R n which supports L, define f K , L ( H )=| K ∩ H |, where |·| denotes the n −1-dimensional Lebesgue measure. We conjecture that K is determined uniquely by f K , L . A number of partial results are presented, and some obvious alternative formulations are shown to be trivial.

On characterizing collections arising from N-gons in the plane

Abstract In this paper we show that for any dimension