Introduction to Unit Balls

We bound the 1-width of a Riemannian manifold in terms of its first homology and the supremal width of its unit balls.

In this paper, we will introduce two new geometric constants JL(X) and YJ(X) in Banach spaces, which are symmetric and related to the side lengths of inscribed equilateral triangles of unit balls.

A classic result of Banach states that the supreme of a multivariate homogenous polynomial is equivalent to that of its associated symmetric multilinear form over unit balls.

For \(n, m\geq 2\), we characterize the smooth points of the unit balls of the spaces \({\mathcal L}(^nl_{\infty}^m)\) and \({\mathcal L}_s(^nl_{\infty}^m).

A prominent subfamily of sphere packings is formed by the so-called totally separable sphere packings: here, a packing of unit balls in Euclidean d-space is called totally separable if any two unit balls can be separated by a hyperplane such that it is disjoint from the interior of each unit ball in the packing.

In this article, we provide a family of polytopes in Z^2g which can be realized as dual unit balls of Thurston norms on 3-manifolds.

We provide an explicit formula for $$\Vert \cdot \Vert _{m,n}$$ , a full description of the extreme points of the corresponding unit balls and a parametrization and a plot of their unit spheres for certain values of m and n.

The theory we develop establishes that deep networks are Kolmogorov-optimal approximants for markedly different function classes, such as unit balls in Besov spaces and modulation spaces.

The technique used in all three cases relies on characterizations of the extreme points of the unit balls of these spaces.

We prove the corresponding bounds for the unit balls of the trigonometric polynomials with frequencies from a hyperbolic cross.

We prove an analogue of this theorem for ball polyhedra, that is, for intersections of finitely many unit balls in $$\mathbb {R}^3$$ R 3.

Spaces which are constructed from unit balls in complex Euclidean spaces are called spherical and are very well understood.

The derivation of equations for real order parameters in Kuramoto models on spheres is based on recently unveiled connections of these models with geometries of unit balls.

The same idea applies to develop optimal MIS  algorithm for higher dimensional unit balls and unit hypercubes.

We first examine the performance of the method to compute the volumes of star-convex unit balls and show that it gives accurate estimates of their volumes.

This characterization generalizes results of D'Angelo–Huo–Xiao and D'Angelo–Lebl, where the codomains are the unit balls.

, by Cowen–MacCluer and Bisi–Bracci on the unit balls) to a much more general class of domains, called generalized type I domains , which includes in particular the classical bounded symmetric domains of type I and the generalized balls.

In this paper, we prove that the $$\bar{\partial }$$∂¯-operator has closed range in $$L^p$$Lp-spaces and further the canonical solution of the $$\bar{\partial }$$∂¯-problem gains 1/2 derivative in the so-called partial $$L^p$$Lp-Sobolev spaces as well as global boundary regularity for $$\bar{\partial }$$∂¯ on products of unit balls.

We say that X has the intersection property ( I ) (general intersection property (GI), respectively) if, for each countable family (for each family, respectively) { B i } i ∈ A of equivalent closed unit balls such that B X = ⋂ i ∈ A B i , one has B X ⁎ ⁎ = ⋂ i ∈ A B i ∘ ∘ , where B i ∘ ∘ is the bipolar set of B i , that is, the bidual unit ball corresponding to B i.