Related papers: Uneven Splitting of Ham Sandwiches

For a connected $n$-dimensional compact smooth hypersurface $M$ without boundary embedded in $\mathbb{R}^{n+1}$, a classical result of Aleksandrov shows that it must be a sphere if it has constant mean curvature. Li and Nirenberg studied a…

Let {(Z_i,W_i):i=1,...,n} be uniformly distributed in [0,1]^d * G(k,d), where G(k,d) denotes the space of k-dimensional linear subspaces of R^d. For a differentiable function f from [0,1]^k to [0,1]^d we say that f interpolates (z,w) in…

Given a set $A$ of real numbers consider the complete graph on the elements of $A$. We prove that if $A$ is an arithmetic progression then for every vertex $a\in A$ there exists an hamiltonian path such that the absolute differences of…

In this paper we settle the two-dimensional case of a conjecture involving unknown semialgebraic functions with specified smoothness. More precisely, we prove the following result: Let $\mathcal{H}$ be a semialgebraic bundle with respect to…

In this paper we deal with a scale of reproducing kernel Hilbert spaces $H^{(n)}_2$, $n\ge 0$, which are linear subspaces of the classical Hilbertian Hardy space on the right-hand half-plane $\mathbb{C}^+$. They are obtained as ranges of…

Any arrangement of hyperplanes in general position in $P^n$ can be regarded as a divisor with normal crossing. We study the bundles of logarithmic 1-forms corresponding to such divisors` from the point of view of classification of vector…

Let $H_{a,b}^n$ denote the component of the Hilbert scheme whose general point parameterizes an $a$-plane union a $b$-plane meeting transversely in $\mathbf{P}^n$. We describe the effective and nef cones of $H_{a,b}^n$ and determine when…

Multidimensional persistence has been proposed to study the persistence of topological features in data indexed by multiple parameters. In this work, we further explore its algebraic complications from the point of view of higher…

In this paper we investigate the minimum number of maximal subgroups H_i for i=1 ...k of the symmetric group S_n (or the alternating group A_n) such that each element in the group S_n (respectively A_n) lies in some conjugate of one of the…

We prove that every hyperplane passing through the origin in $\rr^{n+1}$ divides an embedded compact free boundary minimal hypersurface of the euclidean $(n+1)$-ball in exactly two connected hypersurfaces. We also show that if a region in…

Let $(P,E)$ be a $(d+1)$-uniform geometric hypergraph, where $P$ is an $n$-point set in general position in $\mathbb{R}^d$ and $E\subseteq {P\choose d+1}$ is a collection of $\epsilon{n\choose d+1}$ $d$-dimensional simplices with vertices…

Let $f_1,\...,f_r$ be polynomials in $n$ variables over a finite field $F$ of cardinality $q$ and characteristic $p$. Let $f_i$ have total degree $d_i$ and define $d=d_1+\...+d_r$. Write $Z$ for the set of common zeros of the $f_i$, over…

We prove that if $G$ is an abelian group and $H_1x_1,\dots,H_{k}x_k$ is an irredundant (minimal) cover of $G$ with cosets, then $$|G:\bigcap_{i=1}^{k}H_{i}|=2^{O(k)}.$$ This bound is the best possible up to the constant hidden in the…

Let $P$ be a set of $n$ points in real projective $d$-space, not all contained in a hyperplane, such that any $d$ points span a hyperplane. An ordinary hyperplane of $P$ is a hyperplane containing exactly $d$ points of $P$. We show that if…

Let $ V $ a vector space of dimension $n$. A $V$ family $ \{H_1, \ldots, H_p \} $ of vectorial hyperplanes being distinct two by two defines an arrangement $ {\cal A}_p = {\cal A} ( H_1, \ldots ,H_p ) $ of $ V $. For $ i \in \{ 1, \ldots, p…

In the present paper, we study hemi-slant submanifolds of a locally product Riemannian manifold. We prove that the anti-invariant distribution which is involved in the definition of hemi-slant submanifold is integrable and give some…

A hyperplane arrangement is said to satisfy the ``Riemann hypothesis'' if all roots of its characteristic polynomial have the same real part. This property was conjectured by Postnikov and Stanley for certain families of arrangements which…

In this article we prove in main Theorem A that any infinity type real hyperplane arrangement $\mathcal{H}_n^m$ (Definition 2.11) with the associated normal system $\mathcal{N}$ (Definitions [2.2,2.4] can be represented isomorphically…

Let $S$ be a set of $n$ points in $\mathbb{R}^d$, where $d \geq 2$ is a constant, and let $H_1,H_2,\ldots,H_{m+1}$ be a sequence of vertical hyperplanes that are sorted by their first coordinates, such that exactly $n/m$ points of $S$ are…

Let a_1,...,a_k satisfy a_1+...+a_k=1 and suppose a k-uniform hypergraph on n vertices satisfies the following property; in any partition of its vertices into k sets A_1,...,A_k of sizes a_1*n,...,a_k*n, the number of edges intersecting…