Related papers: Spaces with the maximal projection constant revisi…

We investigate projection constants within classes of multivariate polynomials over finite-dimensional real Hilbert spaces. Specifically, we consider the projection constant for spaces of spherical harmonics and spaces of homogeneous…

The maximal hyperplane section of the $l_\infty^n$-ball, i.e. of the $n$-cube, is the one perpendicular to 1/sqrt 2 (1,1,0, ... ,0), as shown by Ball. Eskenazis, Nayar and Tkocz extended this result to the $l_p^n$-balls for very large $p…

A shape optimization program is developed for the ratio of Riesz capacities $\text{Cap}_q(K)/\text{Cap}_p(K)$, where $K$ ranges over compact sets in $\mathbb{R}^n$. In different regions of the $pq$-parameter plane, maximality is conjectured…

The unitary group U(N) acts by conjugations on the space H(N) of NxN Hermitian matrices, and every orbit of this action carries a unique invariant probability measure called an orbital measure. Consider the projection of the space H(N) onto…

This paper contains two results on the dimension and smoothness of radial projections of sets and measures in Euclidean spaces. To introduce the first one, assume that $E,K \subset \mathbb{R}^{2}$ are non-empty Borel sets with…

A double-normal pair of a finite set $S$ of points from $R^d$ is a pair of points $\{p,q\}$ from $S$ such that $S$ lies in the closed strip bounded by the hyperplanes through $p$ and $q$ perpendicular to $pq$. A double-normal pair $pq$ is…

Optimal transport enables one to construct a metric on the set of (sufficiently small at infinity) probability measures on any (not too wild) metric space X, called its Wasserstein space W(X). In this paper we investigate the geometry of…

Finding the maximum cardinality of a $2$-distance set in Euclidean space is a classical problem in geometry. Lison\v{e}k in 1997 constructed a maximum $2$-distance set in $\mathbb R^8$ with $45$ points. That $2$-distance set constructed by…

We prove that, in Minkowski space, if a spacelike, $(n-1)$-convex hypersurface $M$ with constant $\sigma_{n-1}$ curvature has bounded principal curvatures, then $M$ is convex. Moreover, if $M$ is not strictly convex, after an…

For $d\geq 2$, we discuss $d$-dimensional complex manifolds $M$ that are the increasing union of bounded open sets $M_n$'s of $\mathbb{C}^d$ with a common uniform squeezing constant. The description of $M$ is given in terms of the corank of…

Among probability measures on $d$-dimensional real projective space, one which maximizes the expected angle $\arccos(\frac{x}{|x|}\cdot \frac{y}{|y|})$ between independently drawn projective points $x$ and $y$ was conjectured to…

Given an Euclidean space, this paper elucidates the topological link between the partial derivatives of the Minkowski functional associated to a set (assumed to be compact, convex, with a differentiable boundary and a non-empty interior)…

We prove a large deviations principle for orthogonal projections of the unit ball $\mathbb{B}_p^n$ of $\ell_p^n$ onto a random $k$-dimensional linear subspace of $\mathbb{R}^n$ as $n\to\infty$ in the case $2<p\le \infty$ and for the…

Let $M$ be a perfect matching on a set of points in the plane where every edge is a line segment between two points. We say that $M$ is globally maximum if it is a maximum-length matching on all points. We say that $M$ is $k$-local maximum…

A maximal arc of degree k in a finite projective plane P of order q = ks is a set of (q-s+1)k points that meets every line of P in either k or 0 points. The collection of the nonempty intersections of a maximal arc with the lines of P is a…

We call a family $\{Y_1,\dots,Y_I\}$ in Euclidean space an equidistance spacing if $\|y_i - y_j\| = 1$ whenever $y_i \in Y_i, y_j \in Y_j$ and $i \neq j$. In other words, choosing a representative from each set produces a complete distance…

A fake weighted projective space X is a Q-factorial toric variety with Picard number one. As with weighted projective space, X comes equipped with a set of weights (\lambda_0,...,\lambda_n). We see how the singularities of…

It is shown that if every projective set of reals is Lebesgue measurable and has the property of Baire, if every projective set in the plane has a projective uniformization, and if Steel's K exists, then J^K_{\omega_1} \models "there are…

A bounded subset of a normed linear space is said to be (diametrically) complete if it cannot be enlarged without increasing the diameter. A complete super set of a bounded set $K$ having the same diameter as $K$ is called a completion of…

In constant curvatures spaces, there are a lot of characterizations of geodesic balls as optimal domain for shape optimization problems. Although it is natural to expect similar characterizations in rank one symmetric spaces, very few is…