Related papers: Polarization problem on a higher-dimensional spher…
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.
For fixed $d\geq 3$, we construct subsets of the $d$-dimensional lattice cube $[n]^d$ of size $n^{\frac{3}{d + 1} - o(1)}$ with no $d+2$ points on a sphere or a hyperplane. This improves the previously best known bound of…
In this paper, we study the two natural polarizations, namely the standard polarization and the box polarization, of the d-th power of the maximal ideal in a polynomial ring. We show that these polarizations correspond to smooth points in…
We consider the minimal discrete and continuous energy problems on the unit sphere $\mathbb{S}^d$ in the Euclidean space $\mathbb{R}^{d+1}$ in the presence of an external field due to finitely many localized charge distributions on…
The standard simplex in R^n, also known as the probability simplex, is the set of nonnegative vectors whose entries sum up to 1. They frequently appear as constraints in optimization problems that arise in machine learning, statistics, data…
In the maximum independent set of convex polygons problem, we are given a set of $n$ convex polygons in the plane with the objective of selecting a maximum cardinality subset of non-overlapping polygons. Here we study a special case of the…
We enumerate and classify all stationary logarithmic configurations of d+2 points on the unit (d-1)-sphere in d-dimensions. In particular, we show that the logarithmic energy attains its relative minima at configurations that consist of two…
Every graph $G$ can be represented by a collection of equi-radii spheres in a $d$-dimensional metric $\Delta$ such that there is an edge $uv$ in $G$ if and only if the spheres corresponding to $u$ and $v$ intersect. The smallest integer $d$…
In terms of the minimal $N$-point diameter $D_d(N)$ for $R^d,$ we determine, for a class of continuous real-valued functions $f$ on $[0,+\infty],$ the $N$-point $f$-best-packing constant $\min\{f(\|x-y\|)\, :\, x,y\in \R^d\}$, where the…
Let $\mathscr{X}$ be the boundary complex of a $(d+1)$-polytope, and let $\rho(d+1,k) = \frac{1}{2}[{\lceil (d+1)/2 \rceil \choose d-k} + {\lfloor (d+1)/2 \rfloor \choose d-k}]$. Recently, the author, answering B\'ar\'any's question from…
An ordinary hypersphere of a set of points in real $d$-space, where no $d+1$ points lie on a $(d-2)$-sphere or a $(d-2)$-flat, is a hypersphere (including the degenerate case of a hyperplane) that contains exactly $d+1$ points of the set.…
We give sufficient conditions on initial and target measures supported on the sphere $\S^n$ to ensure the solution to the optimal transport problem with the cost $|x-y|^2/2$ is a diffeomorphism.
In this paper we first make and discuss a conjecture concerning Newtonian potentials in Euclidean n space which have all their mass on the unit sphere about the origin, and are normalized to be one at the origin. The conjecture essentially…
The motion on the sphere $S^2$ with the potential $V= (x_1x_2x_3)^{-2/3}$ is considered. The Lax representation and the linearisation procedure for this two-dimensional integrable system are discussed.
We consider the inverse conductivity problem with discontinuous conductivities. We show in a rigorous way, by a convergence analysis, that one can construct a completely discrete minimization problem whose solution is a good approximation…
The colourful simplicial depth problem in dimension d is to find a configuration of (d+1) sets of (d+1) points such that the origin is contained in the convex hull of each set (colour) but contained in a minimal number of colourful…
Given a sphere of any radius $r$ in an $n$-dimensional Euclidean space, we study the coverings of this sphere with solid spheres of radius one. Our goal is to design a covering of the lowest covering density, which defines the average…
We study the min-max optimization problem where each function contributing to the max operation is strongly-convex and smooth with bounded gradient in the search domain. By smoothing the max operator, we show the ability to achieve an…
We derive universal lower and upper bounds for max-min and min-max problems (also known as polarization) for the potential of spherical $(k,k)$-designs and provide certain examples, including unit-norm tight frames, that attain these…
We study the problem of maximizing the geometric mean of $d$ low-degree non-negative forms on the real or complex sphere in $n$ variables. We show that this highly non-convex problem is NP-hard even when the forms are quadratic and is…