Related papers: The geometry of $L_0$

Machine learning often aims to produce latent embeddings of inputs which lie in a larger, abstract mathematical space. For example, in the field of 3D modeling, subsets of Euclidean space can be embedded as vectors using implicit neural…

A new intrinsic volume metric is introduced for the class of convex bodies in $\mathbb{R}^n$. As an application, an inequality is proved for the asymptotic best approximation of the Euclidean unit ball by arbitrarily positioned polytopes…

It is shown that any smooth closed orientable manifold of dimension $2k + 1$, $k \geq 2$, admits a smooth polynomially convex embedding into $\mathbb C^{3k}$. This improves by $1$ the previously known lower bound of $3k+1$ on the possible…

The notion of ball convexity, considered in finite dimensional real Banach spaces, is a natural and useful extension of usual convexity; one replaces intersections of half-spaces by suitable intersections of balls. A subset $S$ of a normed…

We use the intrinsic area to define a distance on the space of homothety classes of convex bodies in the $n$-dimensional Euclidean space, which makes it isometric to a convex subset of the infinite dimensional hyperbolic space. The ambient…

This paper introduces a natural definition for the volume of the unit ball in $n$-dimensional normed spaces $\mathbb{R}^n$. This definition preserves the Euclidean relation $P(B)/V(B)=n$ between the perimiter and the volume of the unit ball…

Intrinsic $L_p$ metrics are defined and shown to satisfy a dimension--free bound with respect to the Hausdorff metric.

Definition. A symmetric with respect to 0 bounded closed convex set A in a finite dimensional normed space X is called a sufficient enlargement for X (or of B(X)) if for arbitrary isometric embedding of X into a Banach space Y there exists…

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…

Given an open set with finite perimeter $\Omega\subset \mathbb{R}^n$, we consider the space $LD_\gamma^{p}(\Omega)$, $1\leq p<\infty$, of functions with $p$th-integrable deformation tensor on $\Omega$ and with $p$ th-integrable trace value…

We study relations of some classes of $k$-convex, $k$-visible bodies in Euclidean spaces. We introduce and study \textrm{circular projections} in normed linear spaces and classes of bodies related with families of such maps, in particular,…

We evaluate the shattering dimension of various classes of linear functionals on various symmetric convex sets. The proofs here relay mostly on methods from the local theory of normed spaces and include volume estimates, factorization…

In this paper, we present a method of embedding physics data manifolds with metric structure into lower dimensional spaces with simpler metrics, such as Euclidean and Hyperbolic spaces. We then demonstrate that it can be a powerful step in…

The celebrated Dvoretzky theorem asserts that every $N$-dimensional convex body admits central sections of dimension $d = \Omega(\log N)$, which is nearly spherical. For many instances of convex bodies, typically unit balls with respect to…

We suggest a concept of generalized `angles' in arbitrary real normed vector spaces. We give for each real number a definition of an `angle' by means of the shape of the unit ball. They all yield the well known Euclidean angle in the…

This article derives closed-form parametric formulas for the Minkowski sums of convex bodies in d-dimensional Euclidean space with boundaries that are smooth and have all positive sectional curvatures at every point. Under these conditions,…

We study the symplectic geometry of the moduli space of closed n-gons with fixed side-lengths in hyperbolic 3-space. We prove that these moduli spaces have a symplectic structure coming from Poisson Lie theory. We construct completely…

The space ${\Bbb{L}}$ of oriented lines, or rays, in ${\Bbb{R}}^3$ is a 4-dimensional space with an abundance of natural geometric structure. In particular, it boasts a neutral K\"ahler metric which is closely related to the Euclidean…

The symmetries of surfaces which can be embedded into the symmetries of the 3-dimensional Euclidean space $\mathbb{R}^3$ are easier to feel by human's intuition. We give the maximum order of finite group actions on $(\mathbb{R}^3, \Sigma)$…

For a family $\mathcal{C}$ of properly embedded curves in the 2-dimensional disk $\mathbb{D}^{2}$ satisfying certain uniqueness properties, we consider convex polygons $P\subset \mathbb{D}^{2}$ and define a metric $d$ on $P$ such that…