Regularity properties of spheres in homogeneous groups

Enrico Le Donne; Sebastiano Nicolussi Golo

Home/Archive/arXiv

arXiv · 1509.03881

Regularity properties of spheres in homogeneous groups

Enrico Le Donne·Sebastiano Nicolussi Golo

View original source ↗Read source document ↗Export metadata ↓

Abstract

We study left-invariant distances on Lie groups for which there exists a one-parameter family of homothetic automorphisms. The main examples are Carnot groups, in particular the Heisenberg group with the standard dilations. We are interested in criteria implying that, locally and away from the diagonal, the distance is Euclidean Lipschitz and, consequently, that the metric spheres are boundaries of Lipschitz domains in the Euclidean sense. In the first part of the paper, we consider geodesic distances. In this case, we actually prove the regularity of the distance in the more general context of sub-Finsler manifolds with no abnormal geodesics. Secondly, for general groups we identify an algebraic criterium in terms of the dilating automorphisms, which for example makes us conclude the regularity of homogeneous distances on the Heisenberg group.In such a group, we analyze in more details the geometry of metric spheres. We also provide examples of homogeneous groups where spheres presents cusps.

Explore related subjects

math.MG math.DG math.GR

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Enrico Le Donne, Sebastiano Nicolussi Golo. 2015-09-13. Regularity properties of spheres in homogeneous groups. https://arxiv.org/abs/1509.03881

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

From the Steiner Inellipse to the John Ellipsoid of a Simplex: A Corner-Volume Characterization

For a triangle of area $T$, a planar corner-area characterization states that an interior point $M$ lies on the Steiner inellipse precisely when the three corner triangles cut off by the lines through $M$ parallel to the sides have areas $T_1,T_2,T_3$ satisfying $$ T_1+T_2+T_3=\frac12 T. $$ We give the corresponding statement for a simplex in arbitrary dimension. If $S$ is a nondegenerate $n$-simplex of volume $V$ and $V_1(M),\ldots,V_{n+1}(M)$ are the volumes of the facet-parallel corner simplices determined by $M$, then $$ M\in\partial E_J(S) \quad\Longleftrightarrow\quad \sum_{i=1}^{n+1}V_i(M)^{2/n}=\frac1n V^{2/n}, $$ where $E_J(S)$ is the John ellipsoid of $S$. We also identify the entire corner-volume functional with the central second-moment quadratic of the uniform simplex. The novelty claimed here is limited to the corner-volume formulations and their connections with the planar Steiner-inellipse result; the underlying barycentric, covariance, and John-ellipsoid facts are classical.

math.MG↗

arXiv2026

The $L_p$ Minkowski problem for $C$-close sets: existence and continuity

Let $C$ be a pointed closed convex cone in $\mathbb{R}^n$ with nonempty interior, and let $S^{n-1}$ denote the unit sphere in $\mathbb{R}^n$. The $L_p$ Minkowski problem for $C$-close sets is to determine, for a real number $p$ and a nonzero finite Borel measure $μ$ defined on $Ω_{C^\circ}=S^{n-1}\cap \mathrm{int} C^{\circ}$, whether there exists a $C$-close set $\mathds{A}$ such that $μ$ is the $L_p$ surface area measure of $\mathds{A}$. In this paper, we will solve the problem for $p\in (0,1)$ and for $μ$ being a nonzero finite Borel measure on $Ω_{C^\circ}$. Moreover, we establish the continuity of solutions to the $L_p$ Minkowski problem for $p\in [0, 1]$ in several settings.

math.MG↗

arXiv2026

Convergence of metric measure spaces via embeddings in the Urysohn universal space

We study different notions of convergence of metric measure spaces by means of isometric embeddings into the Urysohn universal metric space $\mathbb U$. Due to the universality of $\mathbb U$, the collection $\mathbb X_1$ of isomorphism classes of normalised metric measure spaces can be canonically identified with the quotient (set) $\mathscr P_\sim(\mathbb U)=\mathscr P(\mathbb U)/\sim$ of the space $\mathscr P(\mathbb U)$ of Borel probability measures on $\mathbb U$, where $μ\simν$ if $ν$ is the pushforward of $μ$ under an isometry between their respective supports. By making crucial use of the ultrahomogeneity of $\mathbb U$, we show that, under the above identification, Gromov's box topology on $\mathbb X_1$ coincides with the quotient topology induced by the weak topology of $\mathscr P(\mathbb U)$. More quantitatively, the truncated $1$-Wasserstein distance on $\mathscr P(\mathbb U)$ induces a complete and separable distance ${\sf d}_{\rm mG}$ on $\mathbb X_1\cong\mathscr P_\sim(\mathbb U)$, which metrises the quotient topology of $\mathscr P_\sim(\mathbb U)$ and is Hölder equivalent to the box distance $\square$.

math.MG↗