Related papers: Geometric constructibility of polygons lying on a …
It is known that the paving conjecture fails for 2-paving projections with constant diagonal 1/2. But the proofs of this fact are existence proofs. We will give concrete examples of these projections and projections with constant diagonal…
We prove the following comparison theorem for metrics with nonnegative scalar curvature, also known as the dihedral rigidity conjecture by Gromov: for $n\le 7$, if an $n$-dimensional prism has nonnegative scalar curvature and weakly mean…
We construct, for any positive integer n, a family of n congruent convex polyhedra in R^3, such that every pair intersects in a common facet. Previously, the largest such family contained only eight polytopes. Our polyhedra are Voronoi…
In the $2$-dimensional $n$-body problem, $n\ge 3$, in spaces of constant curvature, $\kappa\ne 0$, we study polygonal homographic solutions. We first provide necessary and sufficient conditions for the existence of these orbits and then…
It is unknown whether every polycube (polyhedron constructed by gluing cubes face-to-face) has an edge unfolding, that is, cuts along edges of the cubes that unfolds the polycube to a single nonoverlapping polygon in the plane. Here we…
A simple $n$-gon is a polygon with $n$ edges with each vertex belonging to exactly two edges and every other point belonging to at most one edge. Brass asked the following question: For $n \geq 5$ odd, what is the maximum perimeter of a…
On a regular tetrahedron in spherical space there exist the finite number of simple closed geodesics. For any pair of coprime integers $(p,q)$ it was found the numbers $\alpha_1$ and $\alpha_2$ depending on $p$, $q$ and satisfying the…
In this work we study inside-out dissections of polygons and polyhedra. We first show that an arbitrary polygon can be inside-out dissected with $2n+1$ pieces, thereby improving the best previous upper bound of $4(n-2)$ pieces.…
In this paper we answer a question posed by V.V. Batyrev. The question asked if there exists a complete regular fan with more than quadratically many primitive collections. We construct a smooth projective toric variety associated to a…
A point $p$ is said to be an area center of a polygon if all of the triangles composed of $p$ and its edges have one and the same area. We construct a moduli space $AC_n$ of such $n$-gons and study its geometry and arithmetic. For every…
We prove that a simplicial 2-sphere satisfying a certain condition is the underlying simplicial complex of a 3-dimensional non-singular complete fan. In particular, this implies that any simplicial 2-sphere with $\leq 18$ vertices is the…
A small polygon is a polygon of unit diameter. The maximal area of a small polygon with $n=2m$ vertices is not known when $m \ge 7$. In this paper, we construct, for each $n=2m$ and $m\ge 3$, a small $n$-gon whose area is the maximal value…
A quasigeodesic is a curve on the surface of a convex polyhedron that has $\le \pi$ surface to each side at every point. In contrast, a geodesic has exactly $\pi$ to each side and so can never pass through a vertex, whereas quasigeodesics…
One-dimensional sigma-models with N supersymmetries are considered. For conventional supersymmetries there must be N-1 complex structures satisfying a Clifford algebra and the constraints on the target space geometry can be formulated in…
We construct Fano threefolds with very ample anti-canonical bundle and Picard rank greater than one from cracked polytopes - polytopes whose intersection with a complete fan forms a set of unimodular polytopes - using Laurent inversion; a…
It is an important question whether it is possible to put a geometry on a given manifold or not. It is well known that any simply connected closed manifold admitting a real projective structure must be a sphere. Therefore, any simply…
We explore optimal circular nonconvex partitions of regular k-gons. The circularity of a polygon is measured by its aspect ratio: the ratio of the radii of the smallest circumscribing circle to the largest inscribed disk. An optimal…
The content of this paper is a generalization of a the theorem 9.2 of the paper arXiv:1007.2665 written by Joseph Rabinoff : if $\mathcal{P}$ is a finite family of polyhedra in $N_{\mathbb{R}}$ such that there exists a fan in…
The study of comparison theorems in geometry has a rich history. In this paper, we establish a comparison theorem for polyhedra in 3-manifolds with nonnegative scalar curvature, answering affirmatively a dihedral rigidity conjecture by…
The main result of this paper is a proof that a nearly flat, acutely triangulated convex cap C in R^3 has an edge-unfolding to a non-overlapping polygon in the plane. A convex cap is the intersection of the surface of a convex polyhedron…