We consider the question of the largest possible combinatorial diameter among pure dimensional and strongly connected (d-1)-dimensional simplicial complexes on n vertices, denoted H_s(n, d). Using a probabilistic construction we give a new lower bound on H_s(n, d) that is within an O(d^2) factor of the upper bound. This improves on the previously best known lower bound which was within a factor of e^{varTheta (d)} of the upper bound. We also make a similar improvement in the case of pseudomanifolds.

Convex polygons are distinguishable among the piecewise C^infty convex domains by comparing their visual angle functions on any surrounding circle. This is a consequence of our main result, that every segment in a C^infty multicurve can be reconstructed from the masking function of the multicurve given on any surrounding circle.

We show that a convex body admits a translative dense packing in mathbb {R}^d if and only if it admits a translative economical covering.

We define a general framework that includes objects such as tilings, Delone sets, functions, and measures. We define local derivability and mutual local derivability (MLD) between any two of these objects in order to describe their interrelation. This is a generalization of the local derivability and MLD (or S-MLD) for tilings and Delone sets which are used in literature, under a mild assumption. We show that several canonical maps in aperiodic order send an object mathcal {P} to one that is MLD with mathcal {P}. Moreover, we show that, for an object mathcal {P} and a class Sigma of objects, a mild condition on them ensures that there exists some mathcal {Q}in Sigma that is MLD with mathcal {P}. As an application, we study pattern-equivariant functions. In particular, we show that the space of all pattern-equivariant functions contains all the information on the original object up to MLD, in a quite general setting.

An ordinary circle of a set P of n points in the plane is defined as a circle that contains exactly three points of P. We show that if P is not contained in a line or a circle, then P spans at least n^2/4 - O(n) ordinary circles. Moreover, we determine the exact minimum number of ordinary circles for all sufficiently large n and describe all point sets that come close to this minimum. We also consider the circle variant of the orchard problem. We prove that P spans at most n^3/24 - O(n^2) circles passing through exactly four points of P. Here we determine the exact maximum and the extremal configurations for all sufficiently large n. These results are based on the following structure theorem. If n is sufficiently large depending on K, and P is a set of n points spanning at most Kn^2 ordinary circles, then all but O(K) points of P lie on an algebraic curve of degree at most four. Our proofs rely on a recent result of Green and Tao on ordinary lines, combined with circular inversion and some classical results regarding algebraic curves.

Let S be a subset of mathbb {R}^d with finite positive Lebesgue measure. The Beer index of convexity{text {b}}(S) of S is the probability that two points of S chosen uniformly independently at random see each other in S. The convexity ratio{text {c}}(S) of S is the Lebesgue measure of the largest convex subset of S divided by the Lebesgue measure of S. We investigate the relationship between these two natural measures of convexity. We show that every set Ssubseteq mathbb {R}^2 with simply connected components satisfies {text {b}}(S)leqslant alpha {text {c}}(S) for an absolute constant alpha , provided {text {b}}(S) is defined. This implies an affirmative answer to the conjecture of Cabello et al. that this estimate holds for simple polygons. We also consider higher-order generalizations of {text {b}}(S). For 1leqslant kleqslant d, the k-index of convexity{text {b}}_k(S) of a set Ssubseteq mathbb {R}^d is the probability that the convex hull of a (k+1)-tuple of points chosen uniformly independently at random from S is contained in S. We show that for every dgeqslant 2 there is a constant beta (d)>0 such that every set Ssubseteq mathbb {R}^d satisfies {text {b}}_d(S)leqslant beta {text {c}}(S), provided {text {b}}_d(S) exists. We provide an almost matching lower bound by showing that there is a constant gamma (d)>0 such that for every varepsilon in (0,1) there is a set Ssubseteq mathbb {R}^d of Lebesgue measure 1 satisfying {text {c}}(S)leqslant varepsilon and {text {b}}_d(S)geqslant gamma frac{varepsilon }{log _2{1/varepsilon }}geqslant gamma frac{{text {c}}(S)}{log _2{1/{text {c}}(S)}}.

A result of Eckhoff implies that to every finite T(3)-family of pairwise disjoint copies of a closed disc of unit diameter there exists a strip of width 1 meeting all members of the family. Our goal is to generalize this result giving a stricter upper bound by proving that the narrowest transversal strip has width < 0.65.

Consider the d-dimensional Euclidean space . Two main results are presented: First, for any N ∈ ℕ, the number of types of periodic equivariant tilings (T, Γ) that have precisely N orbits of (2,4, 6,...)-flags with respect to the symmetry group Γ, is finite. Second, for any N ∈ ℕ, the number of types of convex, periodic equivariant tilings (T, Γ) that have precisely N orbits of tiles with respect to the symmetry group T, is finite. The former result (and some generalizations) is proved combinatorially, using Delaney symbols, whereas the proof of the latter result is based on both geometric arguments and Delaney symbols.

We construct, for everyd>-4, ad-polytopeP⊂ℝd, containing two verticesv andw such that no hyperplane of ℝd containingv andw has more than one facet ofP in either of its closed half-spaces. The result shows that facet-reducing cuts containing a prespecified pair of vertices of a polytope, do not exist in general.