Related papers: Equipartitions with Wedges and Cones

Hadwiger's theorem is a Helly-type theorem involving common transversals to families of convex sets instead of common intersections. Subsequently, Pollack and Wenger identified a necessary and sufficient condition, called a consistent…

A k-fan is a set of k half-lines (rays) all starting from the same point, called the origin of the fan. We discuss the partition of convex 2D regions into n (a positive integer) equal area convex pieces by fans with the following additional…

Many results in mass partitions are proved by lifting $\mathbb{R}^d$ to a higher-dimensional space and dividing the higher-dimensional space into pieces. We extend such methods to use lifting arguments to polyhedral surfaces. Among other…

An arrangement of k-semilines in the Euclidean (projective) plane or on the 2-sphere is called a k-fan if all semilines start from the same point. A k-fan is an $\alpha$-partition for a probability measure $\mu$ if $\mu(\sigma_i)=\alpha_i$…

A balanced partition is a clustering of a graph into a given number of equal-sized parts. For instance, the Bisection problem asks to remove at most k edges in order to partition the vertices into two equal-sized parts. We prove that…

We prove that the number of partitions of the hypercube ${\bf Z}_q^n$ into $q^m$ subcubes of dimension $n-m$ each for fixed $q$, $m$ and growing $n$ is asymptotically equal to $n^{(q^m-1)/(q-1)}$. For the proof, we introduce the operation…

We introduce and prove the $n$-dimensional Pizza Theorem: Let $\mathcal{H}$ be a hyperplane arrangement in $\mathbb{R}^{n}$. If $K$ is a measurable set of finite volume, the {pizza quantity} of $K$ is the alternating sum of the volumes of…

We prove that any finite collection of polygons of equal area has a common hinged dissection. That is, for any such collection of polygons there exists a chain of polygons hinged at vertices that can be folded in the plane continuously…

Gr\"unbaum's equipartition problem asked if for any measure $\mu$ on $\mathbb{R}^d$ there are always $d$ hyperplanes which divide $\mathbb{R}^d$ into $2^d$ $\mu$-equal parts. This problem is known to have a positive answer for $d\le 3$ and…

We introduce elliptic weights of boxed plane partitions and prove that they give rise to a generalization of MacMahon's product formula for the number of plane partitions in a box. We then focus on the most general positive degenerations of…

We apply supersymmetric localization to N=(2,2) gauged linear sigma models on a hemisphere, with boundary conditions, i.e., D-branes, preserving B-type supersymmetries. We explain how to compute the hemisphere partition function for each…

A generalization of the Borsuk-Ulam theorem to Stiefel manifolds is considered. This theorem is applied to derive bounds on $d$ that guarantee-for a given set of $m$ measures in $\mathbb{R}^d$-the existence of $k$ mutually orthogonal…

We use the measurable Hall's theorem due to Cie\'sla and Sabok to prove that (i) if two measurable sets $A,B \subset \mathbb{R}^d$ of the same measure are bounded remainder sets with respect to a given irrational $d$-dimensional vector…

In this paper, we study the embedding of a complete balanced $d$-partite $d$-uniform hypergraph with all its $nd$ vertices represented as points in general position in $\mathbb{R}^d$ and each hyperedge drawn as a convex hull of $d$…

We study some measure partition problems: Cut the same positive fraction of $d+1$ measures in $\mathbb R^d$ with a hyperplane or find a convex subset of $\mathbb R^d$ on which $d+1$ given measures have the same prescribed value. For both…

Denote by Q_d the d-dimensional hypercube. Addressing a recent question we estimate the number of ways the vertex set of Q_d can be partitioned into vertex disjoint smaller cubes. Among other results, we prove that the asymptotic order of…

Volume-fraction expressions are obtained for the systems of an infinite number of parallel planes arranged both regularly and randomly. As a special case of random arrangement, a non-Poissonian point process (the second-order Erlang…

Rounding has proven to be a fundamental tool in theoretical computer science. By observing that rounding and partitioning of $\mathbb{R}^d$ are equivalent, we introduce the following natural partition problem which we call the {\em secluded…

A union of an arrangement of affine hyperplanes $H$ in $R^d$ is the real algebraic variety associated to the principal ideal generated by the polynomial $p_{H}$ given as the product of the degree one polynomials which define the hyperplanes…

Let $P \subset \mathbb{R}^{d}$ be a closed convex cone. Assume that $P$ is pointed, i.e. the intersection $P \cap -P=\{0\}$ and $P$ is spanning, i.e. $P-P=\mathbb{R}^{d}$. Denote the interior of $P$ by $\Omega$. Let $E$ be a product system…