Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2607.08612

Point-dimension theory (part II): The point-cross dimension

Nadir Maaroufi
View original source ↗Read source document ↗Export metadata ↓

Abstract

We introduce the Point-Cross Dimension, a new pointwise invariant designed to measure the directional organization of a set at a single point. Whereas the Point-Extended Box Dimension quantifies local dispersion and covering complexity, the Point-Cross Dimension isolates a complementary layer: the coexistence of independent effective directions through the same germ. The construction assigns weights to admissible directional probes and aggregates them over projectively independent channels, thereby turning the elementary intuition of a cross into a flexible local dimension theory. This viewpoint separates phenomena that classical isotropic dimensions often collapse. A point may have small local box dispersion while carrying several independent directional channels. Conversely, large local covering complexity need not reflect genuine directional independence. We develop the theory in three successive layers. The first is a point-vector dimension, which records exact local directions. The second is a point-tangential dimension, which replaces exact directions by Bouligand effective directions. The third is the Point-Cross Dimension, which weights these effective projective channels by the point-extended box complexity detected along admissible probes. We establish the basic structural properties of these invariants and compute the resulting Point-Cross Dimension on a range of model configurations, including finite crosses, fractal coordinate frames, oscillatory germs, self-similar curves, Sierpiński-type carpets, Cantor dusts, and infinite-rank outlook examples. The final part of the paper establishes comparison principles between the directional and dispersive layers of the theory.

Explore related subjects

math.MGmath.CAmath.FAmath.GN

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Nadir Maaroufi. 2026-07-09. Point-dimension theory (part II): The point-cross dimension. https://arxiv.org/abs/2607.08612

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↗