Related papers: Points on Hemispheres

We consider an extremal problem for subsets of high-dimensional spheres that can be thought of as an extension of the classical isoperimetric problem on the sphere. Let $A$ be a subset of the $(m-1)$-dimensional sphere $\mathbb{S}^{m-1}$,…

We give an upper bound on the number of perfect matchings in simple graphs with a given number of vertices and edges. We apply this result to give an upper bound on the number of 2-factors in a directed complete bipartite balanced graph on…

For each composite number $n\ne 2^k$, there does not exist a single connected closed $(n+1)$-manifold such that any smooth, simply-connected, closed $n$-manifold can be topologically flat embedded into it. There is a single connected closed…

For a $C^{1}$ degree two latitude preserving endomorphism $f$ of the 2-sphere, we show that $f$ has $2^{n}$ periodic points.

The Hilbert scheme $S^{[n]}$ of points on an algebraic surface $S$ is a simple example of a moduli space and also a nice (crepant) resolution of singularities of the symmetric power $S^{(n)}$. For many phenomena expected for moduli spaces…

We prove a sharp bound for the remainder term of the number of lattice points inside a ball, when averaging over a compact set of (not necessarily unimodular) lattices, in dimensions two and three. We also prove that such a bound cannot…

We show that any point in the convex hull of each of (d+1) sets of (d+1) points in general position in \R^d is contained in at least (d+1)^2/2 simplices with one vertex from each set. This improves the known lower bounds for all d >= 4.

We provide a sufficient condition for a finite number of closed subspaces of a Hilbert space to be linearly independent and their sum to be closed. Under this condition a formula for the orthogonal projection onto the sum is given. We also…

Consider the following problem: how many collinear triples of points must a transversal of (Z/nZ)^2 have? This question is connected with venerable issues in discrete geometry. We show that the answer, for n prime, is between (n-1)/4 and…

Let $S\subset \C^n$, $n\geq 3$ be a compact connected 2-codimensional submanifold having the following property: there exists a Levi-flat hypersurface whose boundary is $S$, possibly as a current. Our goal is to get examples of such $S$…

This article studies the number of ways of selecting $k$ objects arranged in $p$ circles of sizes $n_1,\ldots,n_p$ such that no two selected ones have less than $s$ objects between them. If $n_i\geq sk+1$ for all $1\leq i \leq p$, this…

We show the existence of systems of n polynomial equations in n variables, with a total of n+k+1 distinct monomial terms, possessing [n/k+1]^k nondegenerate positive solutions. (Here, [x] is the integer part of a positive number x.) This…

We show that there are sets of $n$ points in the plane with $n$ arbitrarily large that contain more than $n^{1.014}$ pairs of points separated by a distance exactly $1$. This improves on very recent work of a team at OpenAI, who proved the…

We show the existence of at least two geometrically distinct closed geodesics on an n-dimensional sphere with a bumpy and non-reversible Finsler metric for n>2.

The following problem was proposed in 2010 by S. Lando. Let $M$ and $N$ be two unions of the same number of disjoint circles in a sphere. Do there always exist two spheres in 3-space such that their intersection is transversal and is a…

As is well-known, given the complex sphere P^1 minus two points, there exist nonconstant holomorphic maps from the plane into this set, the simplest example of which is given by applying the exponential map and then composing with a…

We show that if the entropy of any closed hypersurface is close to that of a round hyper-sphere, then it is close to a round sphere in Hausdorff distance. Generalizing the result of \cite{BW1} to higher dimensions.

Given a set $P$ of $n$ points in $\mathbb{R}^3$, we show that, for any $\varepsilon >0$, there exists an $\varepsilon$-net of $P$ for halfspace ranges, of size $O(1/\varepsilon)$. We give five proofs of this result, which are arguably…

Let (G, *) be a semigroup, D subset of G, and n >= 2 be an integer. We say that (D, *) is an n-closed subset of G if a_1* ... *a_n in D for every a_1, ..., a_n in D. Hence every closed set is a 2-closed set. The concept of n-closed sets…

It is verified that the number of vertices in a $d$-dimensional cubical pseudomanifold is at least $2^{d+1}$. Using Adin's cubical $h$-vector, the generalized lower bound conjecture is established for all cubical 4-spheres, as well as for…