Related papers: Convex Pentagons with Positive Heesch Number

We study the limit shape of successive coronas of a tiling, which models the growth of crystals. We define basic terminologies and discuss the existence and uniqueness of corona limits, and then prove that corona limits are completely…

A planar point set is in convex position precisely when it has a convex polygonization, that is, a polygonization with maximum interior angle measure at most \pi. We can thus talk about the convexity of a set of points in terms of the…

In this paper we prove a theorem that provides an upper bound for the density of packings of congruent copies of a given convex body in $\mathbb{R}^n$; this theorem is a generalization of the linear programming bound for sphere packings. We…

The main motivation here is a question: whether any polyhedron which can be subdivided into convex pieces without adding a vertex, and which has the same vertices as a convex polyhedron, is infinitesimally rigid. We prove that it is indeed…

A new approach to prove weak convergence of random polytopes on the space of compact convex sets is presented. This is used to show that the profile of the rescaled Schl\"afli random cone of a random conical tessellation generated by $n$…

In this paper we deal with edge-to-edge, irreducible decompositions of a centrally symmetric convex $(2k)$-gon into centrally symmetric convex pieces. We prove an upper bound on the number of these decompositions for any value of $k$, and…

We prove that every unit area convex pentagon is contained in a convex quadrilateral of area no greater than $3/\sqrt{5}$, and that every unit area convex hexagon is contained in a convex pentagon of area no greater than $7/6$. Both results…

We obtain a good upper bound on the number of solutions of a diophantine equation arising from a strictly convex sequences of real numbers.

In a recent paper, Cs\"ornyei and Wilson prove that curves in Euclidean space of $\sigma$-finite length have tangents on a set of positive $\mathscr{H}^{1}$-measure. They also show that a higher dimensional analogue of this result is not…

Using an intuition from metric geometry, we prove that any flag and normal simplicial complex satisfies the non-revisiting path conjecture. As a consequence, the diameter of its facet-ridge graph is smaller than the number of vertices minus…

In 2012 Gubeladze (Adv.\ Math.\ 2012) introduced the notion of k-convex-normal polytopes to show that integral polytopes all of whose edges are longer than 4d(d+1) have the integer decomposition property. In the first part of this paper we…

There are exactly eight edge-to-edge tilings of the sphere by congruent equilateral pentagons: three pentagonal subdivision tilings with 12, 24, 60 tiles; four earth map tilings with 16, 20, 24, 24 tiles; and one flip modification of the…

Convex polytopes are convex hulls of point sets in the $n$-dimensional space $\E^n$ that generalize 2-dimensional convex polygons and 3-dimensional convex polyhedra. We concentrate on the class of $n$-dimensional polytopes in $\E^n$ called…

Flat surfaces that correspond to meromorphic $1$-forms or to meromorphic quadratic differentials containing poles of order two and higher are surfaces of infinite area. We classify groups that appear as Veech groups of translation surfaces…

For an $(n\ge 2)$-dimensional real Banach space $E$ with unit ball $E_{\le 1}$ and a topological space $X$ arbitrary elements in $C(X,E_{\le 1})$ are always expressible as linear combinations of at most three functions valued in the unit…

In this paper, we study homogeneous convex foliations on the complex projective plane $\mathbb{P}^2$. A foliation is called convex if all of its leaves, except straight lines, have no inflection points, and such foliations form a Zariski…

In this paper, we study the existence of high-dimensional, closed, smooth manifolds whose rational homotopy type resembles that of a projective plane. Applying rational surgery, the problem can be reduced to finding possible Pontryagin…

We study the equivariant genera of strongly invertible and periodic knots. Our techniques include some new strongly invertible concordance group invariants, Donaldson's theorem, and the g-signature. We find many new examples where the…

We show how to edge-unfold a new class of convex polyhedra, specifically a new class of prismatoids (the convex hull of two parallel convex polygons, called the top and base), by constructing a nonoverlapping "petal unfolding" in two new…

We define an 'oriented convex region' as a convex region with a direction of symmetry. An earlier article had touched upon isosceles triangles, rectangles and ellipses. Here, we examine some more possible oriented containers - semicircles,…