Discrete & Computational Geometry最新文献

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.

For a discrete poset ({mathcal {X}}), McCord proved that the natural map (|{{mathcal {X}}}|rightarrow {{mathcal {X}}}), from the order complex to the poset with the Up topology, is a weak homotopy equivalence. Much later, Živaljević defined the notion of order complex for a topological poset. For a large class of topological posets we prove the analog of McCord’s theorem, namely that the natural map from the order complex to the topological poset with the Up topology is a weak homotopy equivalence. A familiar topological example is the Grassmann poset (mathcal {G}_n(mathbb {{mathbb {R}}})) of proper non-zero linear subspaces of ({mathbb {R}}^{n+1}) partially ordered by inclusion. But our motivation in topological combinatorics is to apply the theorem to posets associated with tropical phased matroids over the tropical phase hyperfield, and in particular to elucidate the tropical version of the MacPhersonian Conjecture. This is explained in Sect. 2.

In this paper, we give a spectral approximation result for the Laplacian on submanifolds of Euclidean spaces with singularities by the (epsilon )-neighborhood graph constructed from random points on the submanifold. Our convergence rate for the eigenvalue of the Laplacian is (Oleft( left( log n/nright) ^{1/(m+2)}right) ), where m and n denote the dimension of the manifold and the sample size, respectively.

For a lattice path (nu ) from the origin to a point (a, b) using steps (E=(1,0)) and (N=(0,1)), we construct an associated flow polytope ({mathcal {F}}_{{widehat{G}}_B(nu )}) arising from an acyclic graph where bidirectional edges are permitted. We show that the flow polytope ({mathcal {F}}_{{widehat{G}}_B(nu )}) admits a subdivision dual to a ((w-1))-simplex, where w is the number of valleys in the path ({overline{nu }} = Enu N). Refinements of this subdivision can be obtained by reductions of a polynomial (P_nu ) in a generalization of Mészáros’ subdivision algebra for acyclic root polytopes where negative roots are allowed. Via an integral equivalence between ({mathcal {F}}_{{widehat{G}}_B(nu )}) and the product of simplices (Delta _atimes Delta _b), we thereby obtain a subdivision algebra for a product of two simplices. As a special case, we give a reduction order for reducing (P_nu ) that yields the cyclic (nu )-Tamari complex of Ceballos, Padrol, and Sarmiento.

Let K be a compact convex set in (mathbb {R}^{2}) and let (mathcal {F}_{1}, mathcal {F}_{2}, mathcal {F}_{3}) be finite families of translates of K such that (A cap B ne emptyset ) for every (A in mathcal {F}_{i}) and (B in mathcal {F}_{j}) with (i ne j). A conjecture by Dol’nikov is that, under these conditions, there is always some (j in { 1,2,3 }) such that (mathcal {F}_{j}) can be pierced by 3 points. In this paper we prove a stronger version of this conjecture when K is a body of constant width or when it is close in Banach-Mazur distance to a disk. We also show that the conjecture is true with 8 piercing points instead of 3. Along the way we prove more general statements both in the plane and in higher dimensions. A related result was given by Martínez-Sandoval, Roldán-Pensado and Rubin. They showed that if (mathcal {F}_{1}, dots , mathcal {F}_{d}) are finite families of convex sets in (mathbb {R}^{d}) such that for every choice of sets (C_{1} in mathcal {F}_{1}, dots , C_{d} in mathcal {F}_{d}) the intersection (bigcap _{i=1}^{d} {C_{i}}) is non-empty, then either there exists (j in { 1,2, dots , n }) such that (mathcal {F}_j) can be pierced by few points or (bigcup _{i=1}^{n} mathcal {F}_{i}) can be crossed by few lines. We give optimal values for the number of piercing points and crossing lines needed when (d=2) and also consider the problem restricted to special families of convex sets.

We prove an equivalence between open questions about the embeddability of the space of persistence diagrams and the space of probability distributions (i.e. Wasserstein space). It is known that for many natural metrics, no coarse embedding of either of these two spaces into Hilbert space exists. Some cases remain open, however. In particular, whether coarse embeddings exist with respect to the p-Wasserstein distance for (1le ple 2) remains an open question for the space of persistence diagrams and for Wasserstein space on the plane. In this paper, we show that embeddability for persistence diagrams is equivalent to embeddability for Wasserstein space on (mathbb {R}^2). When (p > 1), Wasserstein space on (mathbb {R}^2) is snowflake universal (an obstruction to embeddability into any Banach space of non-trivial type) if and only if the space of persistence diagrams is snowflake universal.

In the classical linear degeneracy testing problem, we are given n real numbers and a k-variate linear polynomial F, for some constant k, and have to determine whether there exist k numbers (a_1,ldots ,a_k) from the set such that (F(a_1,ldots ,a_k) = 0). We consider a generalization of this problem in which F is an arbitrary constant-degree polynomial, we are given k sets of n real numbers, and have to determine whether there exists a k-tuple of numbers, one in each set, on which F vanishes. We give the first improvement over the naïve (O^*(n^{k-1})) algorithm for this problem (where the (O^*(cdot )) notation omits subpolynomial factors). We show that the problem can be solved in time (O^*left( n^{k - 2 + frac{4}{k+2}}right) ) for even k and in time (O^*left( n^{k - 2 + frac{4k-8}{k^2-5}}right) ) for odd k in the real RAM model of computation. We also prove that for (k=4), the problem can be solved in time (O^*(n^{2.625})) in the algebraic decision tree model, and for (k=5) it can be solved in time (O^*(n^{3.56})) in the same model, both improving on the above uniform bounds. All our results rely on an algebraic generalization of the standard meet-in-the-middle algorithm for k-SUM, powered by recent algorithmic advances in the polynomial method for semi-algebraic range searching. In fact, our main technical result is much more broadly applicable, as it provides a general tool for detecting incidences and other interactions between points and algebraic surfaces in any dimension. In particular, it yields an efficient algorithm for a general, algebraic version of Hopcroft’s point-line incidence detection problem in any dimension.

We study the limits of the cones of symmetric nonnegative polynomials and symmetric sums of squares, when expressed in power-mean or monomial-mean basis. These limits correspond to forms with stable expression in power-mean polynomials that are globally nonnegative (resp. sums of squares) regardless of the number of variables. We introduce partial symmetry reduction to describe the limit cone of symmetric sums of squares, and reprove a result of Blekherman and Riener (Discrete Comput Geom 65:1–36, 2020) that limits of symmetric nonnegative polynomials and sums of squares agree in degree 4. We use tropicalization of the dual cones, first considered in the context of comparing nonnegative polynomials and sums of squares in Blekherman et al. (Trans Am Math Soc 375(09):6281–6310, 2022), to show differences between cones of symmetric polynomials and sums of squares starting in degree 6, which disproves a conjecture of Blekherman and Riener (Discrete Comput Geom 65:1–36, 2020). For even symmetric nonnegative forms and sums of squares we show that the cones agree up to degree 8, and are different starting with degree 10. We also find, via tropicalization, explicit examples of symmetric forms that are nonnegative but not sums of squares in the limit.