Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.19214

Orthogonal partition by four hyperplanes

Alexey Fakhrutdinov·Platon Khamatov·Oleg R. Musin
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study orthogonal $(4,3)$-equipartitions of finite measures absolutely continuous with respect to Lebesgue measure: every three of four mutually orthogonal affine hyperplanes must divide each measure into eight equal parts. For every number $m$ of measures, we determine the exact dimension threshold of Musin's polynomial criterion and express it in terms of the binary expansion of $m$. The threshold coincides with the geometric lower bound precisely when $m=2^j-1$. Consequently, $Δ^*(2^j-1,4,3)=7\cdot2^{j-1}-2$ for every $j\geq1$.

Explore related subjects

math.MGmath.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Alexey Fakhrutdinov, Platon Khamatov, Oleg R. Musin. 2026-10-13. Orthogonal partition by four hyperplanes. https://arxiv.org/abs/2610.19214

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

KEEP EXPLORING

Related papers

arXiv2026

Rational Area Profiles of Regular Polygons Cut by Two Congruent Diagonals

Two congruent diagonals of a regular hexagon cut it into four regions whose areas are all rational multiples of the area of the hexagon. We determine when this phenomenon occurs in a regular $n$-gon. For every regular polygon cut by two intersecting congruent diagonals, neither of which is a diameter, we classify all unions of the four resulting regions whose areas are rational fractions of the total area. The hexagon is the only configuration in which all four regions are simultaneously rational. One of the two regions on the axis of symmetry is rational in exactly five sporadic configurations, with $n=6,18,30$, whereas the other region and the adjacent two-region unions belong to infinite families with simple geometric explanations. The two lateral regions have rational area ratios in exactly two configurations, with $n=6$ and $n=12$. The proof reduces each area ratio to a trigonometric quotient, bounds its rational values by norms in cyclotomic fields, and settles the remaining equations using Myerson's classification of equal products of two sines and the Poonen--Rubinstein classification of a three-sine-product equation.

math.MG↗
arXiv2026

Stability of Hyperplane Projections of the Regular Simplex and $L_p$-Projection Bodies of balls in $\ell_q^n(\mathbb{R})$

We prove stability estimates for extremal hyperplane projections of the regular simplex. We further generalize the analogous stability estimates for $\ell_q$-balls by the first named author and the monotonicity theorem for normalized ratios by Barthe and Naor to the $L_p$-setting. To this end, we derive closed formulas for the support functions of their $L_p$-projection bodies, complementing those obtained by the second named author. A key ingredient in our proofs is a stability refinement of the majorization inequality for Gaussian mixtures.

math.MG↗
arXiv2026

Non-convex polytopes realizing kissing contact graphs in dimensions 5-7

Each of the 12 known conjecturally optimal kissing configurations in dimensions 5, 6, and 7 carries a contact-graph polytope: a polytope whose vertices are the configuration points and whose edges are exactly its contacts. Eight of these polytopes are non-convex and distinct from their convex hulls, which include longer edges. Each polytope is radial about the sphere's center: its boundary meets every ray from the center exactly once. It is the unique radial polytope with those vertices and edges. Seven of the nine non-lattice configurations carry polytopes constructed from the lattice contact polytope in the same dimension by splitting some facets and reassembling the pieces. This split-and-reassemble construction describes the configurations by facets rather than by layers. The fold angles of the reassembled facets explain the inner products $1/(10-n)$, giving $1/5$, $1/4$, and $1/3$ in dimensions 5, 6, and 7, respectively. The contact graphs of all 12 configurations are infinitesimally rigid. We also prove that $Q_5$ and $R_5$ are infinitesimally jammed, extending the 10 previously established cases to all 12 configurations. All mathematical computations use exact arithmetic.

math.MG↗