Discrete Midpoint Convexity

For a function defined on a convex set in a Euclidean space, midpoint convexity is the property requiring that the value of the function at the midpoint of any line segment is not greater than the average of its values at the endpoints of the line segment. Midpoint convexity is a well-known characterization of ordinary convexity under very mild assumptions. For a function defined on the integer lattice, we consider the analogous notion of discrete midpoint convexity, a discrete version of midpoint convexity where the value of the function at the (possibly noninteger) midpoint is replaced by the average of the function values at the integer round-up and round-down of the midpoint. It is known that discrete midpoint convexity on all line segments with integer endpoints characterizes L$^{\natural}$-convexity, and that it characterizes submodularity if we restrict the endpoints of the line segments to be at $\ell_\infty$-distance one. By considering discrete midpoint convexity for all pairs at $\ell_\infty$-distance equal to two or not smaller than two, we identify new classes of discrete convex functions, called local and global discrete midpoint convex functions, which are strictly between the classes of L$^{\natural}$-convex and integrally convex functions, and are shown to be stable under scaling and addition. Furthermore, a proximity theorem, with the same small proximity bound as that for L$^{\natural}$-convex functions, is established for discrete midpoint convex functions. Relevant examples of classes of local and global discrete midpoint convex functions are provided.Comment: 39 pages, 6 figures, to appear in Mathematics of Operations Researc

Discrete convexity and unimodularity. I

In this paper we develop a theory of convexity for a free Abelian group M (the lattice of integer points), which we call theory of discrete convexity. We characterize those subsets X of the group M that could be call "convex". One property seems indisputable: X should coincide with the set of all integer points of its convex hull co(X) (in the ambient vector space V). However, this is a first approximation to a proper discrete convexity, because such non-intersecting sets need not be separated by a hyperplane. This issue is closely related to the question when the intersection of two integer polyhedra is an integer polyhedron. We show that unimodular systems (or more generally, pure systems) are in one-to-one correspondence with the classes of discrete convexity. For example, the well-known class of g-polymatroids corresponds to the class of discrete convexity associated to the unimodular system A_n:={\pm e_i, e_i-ej} in Z^n.Comment: 26 pages, Late

Discrete Convex Functions on Graphs and Their Algorithmic Applications

The present article is an exposition of a theory of discrete convex functions on certain graph structures, developed by the author in recent years. This theory is a spin-off of discrete convex analysis by Murota, and is motivated by combinatorial dualities in multiflow problems and the complexity classification of facility location problems on graphs. We outline the theory and algorithmic applications in combinatorial optimization problems

On k-Convex Polygons

We introduce a notion of $k$-convexity and explore polygons in the plane that have this property. Polygons which are \mbox{$k$-convex} can be triangulated with fast yet simple algorithms. However, recognizing them in general is a 3SUM-hard problem. We give a characterization of \mbox{$2$-convex} polygons, a particularly interesting class, and show how to recognize them in \mbox{$O(n \log n)$} time. A description of their shape is given as well, which leads to Erd\H{o}s-Szekeres type results regarding subconfigurations of their vertex sets. Finally, we introduce the concept of generalized geometric permutations, and show that their number can be exponential in the number of \mbox{$2$-convex} objects considered.Comment: 23 pages, 19 figure

Combinatorial Space Tiling

The present article studies combinatorial tilings of Euclidean or spherical spaces by polytopes, serving two main purposes: first, to survey some of the main developments in combinatorial space tiling; and second, to highlight some new and some old open problems in this area.Comment: 16 pages; to appear in "Symmetry: Culture and Science

Testing k-monotonicity of a discrete distribution. Application to the estimation of the number of classes in a population

We develop here several goodness-of-fit tests for testing the k-monotonicity of a discrete density, based on the empirical distribution of the observations. Our tests are non-parametric, easy to implement and are proved to be asymptotically of the desired level and consistent. We propose an estimator of the degree of k-monotonicity of the distribution based on the non-parametric goodness-of-fit tests. We apply our work to the estimation of the total number of classes in a population. A large simulation study allows to assess the performances of our procedures.Comment: 32 pages, 8 figure