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Wednesday, 4 November 2026: 4 new papers, 4 cross-lists and 11 replacements in Metric Geometry (math.MG), in arXiv's order, with full abstracts. The same papers as arXiv's own listing, set as a newspaper; every title links to arXiv.
We show that the kissing numbers in dimensions 32, 33, 34, 35, 37, 38 and 39 are at least 352320, 370240, 390980, 415308, 499816, 605322 and 768916, improving the previous bounds 347584, 364736, 384836, 409676, 498024, 591900 and 763668. Except in dimension 38, the configurations are those of Edel, Rains and Sloane with new binary constant-weight codes of weight 8 and minimum distance 8 in the middle layer; each additional word adds 128 points. In dimension 38 we add 6855 antipodal pairs of vectors of the second shell of the Leech lattice to the empty equator of Kravatskiy's 591612-point configuration, where 144 pairs were known; this is an independent set problem in a regular graph of degree 11178 on 8386560 vertices. The codes and lines were found by local search, and all configurations were checked pair by pair in exact integer arithmetic.
A planar crescent configuration consists of $n$ points, no three collinear and no four concyclic, whose distinct distances have multiplicities $1,2,\ldots,n-1$. We prove by exact computation that no such configuration exists for $n=9$. The three pairs carrying the two least frequent distances yield a six-point subset with at most six distances. Rational linear algebra on quartic polynomials separates the possible six-point distance patterns into those with certified finite lists of normalized distance tuples and a residual family. Compatibility of overlapping subsets forces at least one core into the first family, without classifying the residual realizations. The remaining three points are then obtained by rational constructions over the field of the squared core distances. The multiplication matrices used to enumerate the core distances have order at most twelve. Exhaustive label assignments and exact algebraic comparisons exclude all remaining extensions.
A magic square exemplifies balance in inequality: its $n^2$ entries $\{1,\dots,n^2\}$ are distinct, yet every row, column, and main diagonal sums to $\mu_n=n(n^2+1)/2$. Lifting centred grid cells into $\mathbb{R}^3$ as radial vectors $\mathbf{p}_{ij}(\lambda)=(x_j,y_i,\lambda z_{ij})$ and taking their convex hull---the Magic Gem $\mathcal{G}_\lambda(S)$---asks: when does line-sum equality suffice for equilibrium in 3D, and when a nonlinear realization breaks balance, how can an opposing inequality restore it? We test this across all $880$ order-$4$ magic squares in four regimes. For radial spokes from $O$, natural lengths balance identically, while unit spokes break equilibrium on all $832$ non-associative squares in 3D and on all squares in 2D, where the ring-radius ratio $r_2/r_1=\sqrt{5}$ bounds the resultant below by the Pell convergent $9/4\approx\sqrt{5}$. By length--weight duality, balance is restored via tip weights $\mathbf{w}^*>\mathbf{0}$ on $S^2$ or power-law spoke scaling ($r^{\alpha^*}$ in 2D and $(\mathbf{V}_3)_z=0$ in 3D). In 3D rotation, equal point masses make every semi-magic square an axisymmetric top with spherical-top transition $\lambda_{\mathrm{iso}}=(n^2+1)^{-1/2}$, whereas entry masses $m_{ij}=s_{ij}$ expose the quadrupole $Q_{xy}=\sum z_{ij}x_jy_i$, triggering $180^\circ$ Dzhanibekov flips on $824$ squares. Passing to the uniform solid hull $\mathcal{G}_0(S)$ shifts the centroid off $O$ on $829$ squares; re-engaging the opposing core $\mathcal{G}_1(S)$ restores balance via two-density inclusions, nested wireframes, or the $K_{n^2}$ strut network, while all $880$ hulls unfold into non-overlapping nets with $f_0-1$ glue tabs. Finally, $\lambda=1/5$ stabilizes $74.5\%$ of facets on a table, and alternating $\mathcal{G}_\lambda(S)$ with its mirror twin $\mathcal{G}_\lambda(r^2\overline{S})=-\mathcal{G}_\lambda(S)$ locks chiral blocks into a vertical tower.
We give an elementary proof of a tangent-cone identity which might be known to specialists in geometric measure theory. It concerns Lipschitz mappings $g$ from the closed unit ball $B\subset\mathbb{R}^2$ to $\mathbb{R}^2$. Let $x$ be an interior point of differentiability of $g$ at which $Dg(x)$ has rank one. If the upper one-dimensional Hausdorff density of $g(B)$ at $g(x)$ is at most one, then the tangent cone to $g(B)$ at $g(x)$ coincides with the image of $Dg(x)$. This property is used in the proof of rigidity for the four-gradient problem. We provide a detailed, self-contained proof based on differentiability, connectedness and elementary estimates for one-dimensional Hausdorff measure, without using advanced methods from geometric measure theory.
We study an analogue of the permutohedron for the dihedral group, which we call the dihedral permutohedron. We determine the facet structure of the dihedral permutohedron, use the facet structure to construct a triangulation, and determine a nice combinatorial formula for the volume of the dihedral permutohedron. Lastly, we find a family of interesting unimodular affine symmetries of the odd dihedral permutohedron.
We extend Brown's one-variable rigidity theorem to strictly convex functions of two variables under a regularity assumption. Let $K\subset\mathbb{R}^2$ be nonempty, compact, and convex, and let $f$ be real-valued, strictly convex on $K$, and of class $C^2$ on a neighbourhood of $K$. We prove that the generating set $\{x_1,x_2,f\}$ is hyperrigid in $C(K)$.
We deal with the following two pick-up sticks problems: given an integer n greater than or equal to 3, we choose n sticks with independent random lengths from an exponential distribution or Pareto type I distribution, and we are interested in the probability that no three of them can be the sides of a triangle. In case of exponential distribution, this problem has been solved by other authors: the probability in question is equal to $n!/\big((F_3-1)(F_4-1)\cdots(F_{n+2}-1)\big)$, where $(F_i)_{i=1}^\infty$ is the sequence of Fibonacci numbers. We give a new, self-contained and detailed proof for this formula, which uses a different method compared to the earlier ones. In case of Pareto type I distribution, we derive new formulae for the corresponding probability.
We study the distortion of the lower dimension and lower spectrum of sets under quasiconformal mappings. As an application, we verify weighted Hardy--Sobolev inequalities in Euclidean domains arising as quasiconformal images of domains with sufficiently regular boundaries. The regularity in question is phrased in terms of estimates for either the Assouad or the lower dimension of the complement. We also provide dilatation-dependent classification results for certain planar open sets.