Related papers: Rotors in triangles and tethrahedra

In this paper we study the class of polytopes which can be obtained by taking the convex hull of some subset of the points $\{e_i-e_j \ \vert \ i \neq j\} \cup \{\pm e_i\}$ in $\mathbb{R}^n$, where $e_1,\dots,e_n$ is the standard basis of…

If we fix the angles at the vertices of a convex planar $n$-gon, the lengths of its edges must satisfy two linear constraints in order for it to close up. If we also require unit perimeter, our vectors of $n$ edge lengths form a convex…

A spherical polyhedron surface is a triangulated surface obtained by isometric gluing of spherical triangles. For instance, the boundary of a generic convex polytope in the 3-sphere is a spherical polyhedron surface. This paper investigates…

Planar bipartite graphs can be represented as touching graphs of horizontal and vertical segments in $\mathbb{R}^2$. We study a generalization in space: touching graphs of axis-aligned rectangles in $\mathbb{R}^3$, and prove that planar…

The following problem, arising from medical imaging, is addressed: Suppose that $T$ is a known tetrahedron in $\R^3$ with centroid at the origin. Also known is the orthogonal projection $U$ of the vertices of the image $\phi T$ of $T$ under…

Let $P$ be a set of $n$ points in general position on the plane. A set of closed convex polygons with vertices in $P$, and with pairwise disjoint interiors is called a convex decomposition of $P$ if their union is the convex hull of $P$,…

We study two notions. One is that of spindle convexity. A set of circumradius not greater than one is spindle convex if, for any pair of its points, it contains every short circular arc of radius at least one, connecting them. The other…

We study polar orbitopes, i.e. convex hulls of orbits of a polar representation of a compact Lie group. The face structure is studied by means of the gradient momentum map and it is shown that every face is exposed and is again a polar…

If $ABC$ is a given triangle in the plane, $P$ is any point not on the extended sides of $ABC$ or its anticomplementary triangle, $Q$ is the complement of the isotomic conjugate of $P$ with respect to $ABC$, $DEF$ is the cevian triangle of…

Let $\lambda$ be a dominant weight of a finite dimensional simple Lie algebra and $W$ the Weyl group. The convex hull of $W\lambda$ is defined as the weight polytope of $\lambda$. We provide a new proof that there is a natural bijection…

The distance between convex bodies \(K, L \subseteq \R^n\) is defined as \[ d(K,L)= \inf \left\{ \lambda \ge 1: \ L-x \subseteq T (K-y) \subseteq \lambda (L-x) \right\}, \] where the infimum is taken over all \(x,y \in \R^n\) and all…

We classify rotational surfaces in a normed 3-space with rotationally symmetric norm whose principal curvatures satisfy a linear relation.

We analyze polyhedra composed of hexagons and triangles with three faces around each vertex, and their 3-regular planar graphs of edges and vertices, which we call "trihexes". Trihexes are analogous to fullerenes, which are 3-regular planar…

Given a root system $\Phi$ of type $A_n$, $B_n$, $C_n$, or $D_n$ in Euclidean space $E$, let $W$ be the associated Weyl group. For a point $p \in E$ not orthogonal to any of the roots in $\Phi$, we consider the $W$-permutohedron $P_W$,…

We investigate the motions of a bar structure consisting of two congruent tetrahedra, whose edges in their basic position form the face diagonals of a rectangular parallelepiped. The constraint of the motion is that the originally…

We conjecture that a convex polytope is uniquely determined up to isometry by its edge-graph, edge lengths and the collection of distances of its vertices to some arbitrary interior point, across all dimensions and all combinatorial types.…

A polyhedron is pointed if it contains at least one vertex. Every pointed polyhedron P in R^n can be described by an H-representation consisting of half spaces or equivalently by a V-representation consisting of the convex hull of a set of…

We investigate the intersection body of a convex polytope using tools from combinatorics and real algebraic geometry. In particular, we show that the intersection body of a polytope is always a semialgebraic set and provide an algorithm for…

We investigate the rotation sets of open billiards in $\mathbb{R}^N$ for the natural observable related to a starting point of a given billiard trajectory. We prove that the general rotation set is convex and the set of all convex…

Given a convex body $K\subset \mathbb R^2$ we say that a circle $\Omega\subset \text{int} \ K$ is an equipotential circle if every tangent line of $\Omega$ cuts a chord $AB$ in $K$ such that for the contact point $P=\Omega\cap AB$ it holds…