Discrete & Computational Geometry最新文献

A consistent path system in a graph G is an intersection-closed collection of paths, with exactly one path between any two vertices in G. We call G metrizable if every consistent path system in it is the system of geodesic paths defined by assigning some positive lengths to its edges. We show that metrizable graphs are, in essence, subdivisions of a small family of basic graphs with additional compliant edges. In particular, we show that every metrizable graph with 11 vertices or more is outerplanar plus one vertex.

We study the problem of estimating the convex hull of the image (f(X)subset {mathbb {R}}^n) of a compact set (Xsubset {mathbb {R}}^m) with smooth boundary through a smooth function (f:{mathbb {R}}^mrightarrow {mathbb {R}}^n). Assuming that f is a submersion, we derive a new bound on the Hausdorff distance between the convex hull of f(X) and the convex hull of the images (f(x_i)) of M sampled inputs (x_i) on the boundary of X. When applied to the problem of geometric inference from a random sample, our results give error bounds that are tighter and more general than in previous work. We present applications to the problems of robust optimization, of reachability analysis of dynamical systems, and of robust trajectory optimization under bounded uncertainty.

We study some discrete invariants of Newton non-degenerate polynomial maps (f: {mathbb {K}}^n rightarrow {mathbb {K}}^n) defined over an algebraically closed field of Puiseux series ({mathbb {K}}), equipped with a non-trivial valuation. It is known that the set ({mathcal {S}}(f)) of points at which f is not finite forms an algebraic hypersurface in ({mathbb {K}}^n). The coordinate-wise valuation of ({mathcal {S}}(f)cap ({mathbb {K}}^*)^n) is a piecewise-linear object in ({mathbb {R}}^n), which we call the tropical non-properness set of f. We show that the tropical polynomial map corresponding to f has fibers satisfying a particular combinatorial degeneracy condition exactly over points in the tropical non-properness set of f. We then use this description to outline a polyhedral method for computing this set, and to recover the fan dual to the Newton polytope of the set at which a complex polynomial map is not finite. The proofs rely on classical correspondence and structural results from tropical geometry, combined with a new description of ({mathcal {S}}(f)) in terms of multivariate resultants.

We approximate boundaries of convex polytopes (Xsubset {mathbb {R}}^n) by smooth hypersurfaces (Y=Y_varepsilon ) with positive mean curvatures and, by using basic geometric relations between the scalar curvatures of Riemannian manifolds and the mean curvatures of their boundaries, establish lower bound on the dihedral angles of X.

The lattice of noncrossing partitions is well-known for its wide variety of combinatorial appearances and properties. For example, the lattice is rank-symmetric and enumerated by the Catalan numbers. In this article, we introduce a large family of new noncrossing partition lattices with both of these properties, each parametrized by a configuration of n points in the plane.

We study the problem of finding a minimum homology basis, that is, a lightest set of cycles that generates the 1-dimensional homology classes with (mathbb {Z}_2) coefficients in a given simplicial complex K. This problem has been extensively studied in the last few years. For general complexes, the current best deterministic algorithm, by Dey et al. (LATIN 2018: Theoretical Informatics, Springer International Publishing, Cham, 2018), runs in (O(N m^{omega -1} + n m g)) time, where N denotes the total number of simplices in K, m denotes the number of edges in K, n denotes the number of vertices in K, g denotes the rank of the 1-homology group of K, and (omega ) denotes the exponent of matrix multiplication. In this paper, we present three conceptually simple randomized algorithms that compute a minimum homology basis of a general simplicial complex K. The first algorithm runs in (tilde{O}(m^omega )) time, the second algorithm runs in (O(N m^{omega -1})) time and the third algorithm runs in (tilde{O}(N^2,g + N m g{^2} + m g{^3})) time which is nearly quadratic time when (g=O(1)). We also study the problem of finding a minimum cycle basis in an undirected graph G with n vertices and m edges. The best known algorithm for this problem runs in (O(m^omega )) time. Our algorithm, which has a simpler high-level description, but is slightly more expensive, runs in (tilde{O}(m^omega )) time. We also provide a practical implementation of computing the minimum homology basis for general weighted complexes. The implementation is broadly based on the algorithmic ideas described in this paper, differing in its use of practical subroutines. Of these subroutines, the more costly step makes use of a parallel implementation, thus potentially addressing the issue of scale. We compare results against the currently known state of the art implementation (ShortLoop).

A 3-dimensional polytope P is k-equiprojective when the projection of P along any line that is not parallel to a facet of P is a polygon with k vertices. In 1968, Geoffrey Shephard asked for a description of all equiprojective polytopes. It has been shown recently that the number of combinatorial types of k-equiprojective polytopes is at least linear as a function of k. Here, it is shown that there are at least (k^{3k/2+o(k)}) such combinatorial types as k goes to infinity. This relies on the Goodman–Pollack lower bound on the number of order types of point configurations and on new constructions of equiprojective polytopes via Minkowski sums.

Self-similar sets require a separation condition to admit a nice mathematical structure. The classical open set condition (OSC) is difficult to verify. Zerner proved that there is a positive and finite Hausdorff measure for a weaker separation property which is always fulfilled for crystallographic data. Ngai and Wang gave more specific results for a finite type property (FT), and for algebraic data with a real Pisot expansion factor. We show how the algorithmic FT concept of Bandt and Mesing relates to the property of Ngai and Wang. Merits and limitations of the FT algorithm are discussed. Our main result says that FT is always true in the complex plane if the similarity mappings are given by a complex Pisot expansion factor (lambda ) and algebraic integers in the number field generated by (lambda .) This extends the previous results and opens the door to huge classes of separated self-similar sets, with large complexity and an appearance of natural textures. Numerous examples are provided.

In this paper, we study the connected blocks polytope, which, apart from its own merits, can be seen as the generalization of certain connectivity based or Eulerian subgraph polytopes. We provide a complete facet description of this polytope, characterize its edges and show that it is Hirsch. We also show that connected blocks polytopes admit a regular unimodular triangulation by constructing a squarefree Gröbner basis. In addition, we prove that the polytope is Gorenstein of index 2 and that its (h^*)-vector is unimodal.

While studying set function properties of Lebesgue measure, F. Barthe and M. Madiman proved that Lebesgue measure is fractionally superadditive on compact sets in (mathbb {R}^n). In doing this they proved a fractional generalization of the Brunn–Minkowski–Lyusternik (BML) inequality in dimension (n=1). In this paper we will prove the equality conditions for the fractional superadditive volume inequalites for any dimension. The non-trivial equality conditions are as follows. In the one-dimensional case we will show that for a fractional partition ((mathcal {G},beta )) and nonempty sets (A_1,dots ,A_msubseteq mathbb {R}), equality holds iff for each (Sin mathcal {G}), the set (sum _{iin S}A_i) is an interval. In the case of dimension (nge 2) we will show that equality can hold if and only if the set (sum _{i=1}^{m}A_i) has measure 0.