Related papers: Uneven Splitting of Ham Sandwiches

Schwarz showed that when a closed symplectic manifold (M,\om) is symplectically aspherical (i.e. the symplectic form and the first Chern class vanish on \pi_2(M)) then the spectral invariants, which are initially defined on the universal…

A point set $M$ in $m$-dimensional Euclidean space is called an integral point set if all the distances between the elements of $M$ are integers, and $M$ is not situated on an $(m-1)$-dimensional hyperplane. We improve the linear lower…

In hypothesis testing problems the property of strict unbiasedness describes whether a test is able to discriminate, in the sense of a difference in power, between any distribution in the null hypothesis space and any distribution in the…

We give a complete formula for the characteristic polynomial of hyperplane arrangements $\mathcal J_n$ consisting of the hyperplanes $x_i+x_j=1$, $x_k=0$, $x_l=1$, $ 1\leq i, j, k, l\leq n$. The formula is obtained by associating hyperplane…

We give a complete characterization of invariant subspaces for $(M_{z_1}, \ldots, M_{z_n})$ on the Hardy space $H^2(\mathbb{D}^n)$ over the unit polydisc $\mathbb{D}^n$ in $\mathbb{C}^n$, $n >1$. In particular, this yields a complete set of…

In this note, we prove that smooth self-shrinkers in $\Real^{n+1}$, that are entire graphs, are hyperplanes. Previously Ecker and Huisken showed that smooth self-shrinkers, that are entire graphs and have at most polynomial growth, are…

We study hyperplane covering problems for finite grid-like structures in $\mathbb{R}^d$. We call a set $\mathcal{C}$ of points in $\mathbb{R}^2$ a conical grid if the line $y = a_i$ intersects $\mathcal{C}$ in exactly $i$ points, for some…

The problem of existence of adaptive confidence bands for an unknown density $f$ that belongs to a nested scale of H\"{o}lder classes over $\mathbb{R}$ or $[0,1]$ is considered. Whereas honest adaptive inference in this problem is…

A locally compact stable plane of positive topological dimension will be called semiaffine if for every line $L$ and every point $p$ not in $L$ there is at most one line passing through $p$ and disjoint from $L$. We show that then the plane…

Given an undirected unweighted graph $G = (V, E)$ on $n$ vertices and $m$ edges, a subgraph $H\subseteq G$ is a spanner of $G$ with stretch function $f: \mathbb{R}_+ \rightarrow \mathbb{R}_+$, if for every pair $s, t$ of vertices in $V$,…

In this article we prove in the main theorem that, there is a bijection between the isomorphism classes of a certain type of real hyperplane arrangements on the one hand, and the antipodal pairs of convex cones of an associated…

We argue that the following three statements cannot all be true: (i) our vacuum is a type IIB / F-theory vacuum at moderate-to-large $h^{1,1}$, (ii) the $\alpha'$-expansion is controlled via the supergravity approximation, \`a la the KKLT…

For fixed $r\geq 3$ and $n$ divisible by $r$, let ${\mathcal H}={\mathcal H}^r_{n,M}$ be the random $M$-edge $r$-graph on $V=\{1,\ldots ,n\}$; that is, ${\mathcal H}$ is chosen uniformly from the $M$-subsets of ${\mathcal K}:={V \choose r}$…

Let $A$ be any plane set. It is known that a holomorphic motion $h: A \times \mathbb{D} \to \mathbb{C}$ always extends to a holomorphic motion of the entire plane. It was recently shown that any isotopy $h: X \times [0,1] \to \mathbb{C}$,…

In this paper we show the validity, under certain geometric conditions, of Wheeler's thin sandwich conjecture for higher dimensional theories of gravity. We extend the results shown by R. Bartnik and G. Fodor for the 3-dimensional case in…

Let $\mathcal{H}$ be a hyperplane arrangement in $\mathbb{CP}^n$. We define a quadratic form $Q$ on $\mathbb{R}^{\mathcal{H}}$ that is entirely determined by the intersection poset of $\mathcal{H}$. Using the Bogomolov-Gieseker inequality…

We show that, for any prime power n and any convex body K (i.e., a compact convex set with interior) in Rd, there exists a partition of K into n convex sets with equal volumes and equal surface areas. Similar results regarding…

We classify complex hyperplane arrangements $\mathcal A$ whose intersection posets $L(\mathcal A)$ satisfy $L(\mathcal A)=\pi_i^{-1}\circ\pi_i\bigl(L(\mathcal A)\bigr)$ for $i=1,\dots,n$. Here $\pi_i$ denotes the projection from $\mathbb…

We show that if a nonscalar operator on a separable Hilbert space has a nontrivial invariant subspace, then it has also a nontrivial hyperinvariant subspace. Thus the hyperinvariant subspace problem is equivalent to the invariant subspace…

In this paper we study $N_d(k)$ the smallest positive integer such that any nice measure $\mu$ in $\R^d$ can be partitioned in $N_d(k)$ parts of equal measure so that every hyperplane avoids at least $k$ of them. A theorem of Yao and Yao…