Papers

The Koebe-Andreev-Thurston theorem assigns a 3-dimensional hyperbolic reflection group to each combinatorial polyhedron. A natural question is: which of them are arithmetic? In 2016, Kontorovich-Nakamura conjectured that all arithmetic reflection groups obtained in this way are commensurable to those obtained from the tetrahedron, square pyramid, or cuboctahedron. In this paper, we prove the conjecture. It is a consequence of the following result of independent interest: all arithmetic ideal, right-angled hyperbolic polyhedra are obtained by gluing together copies of one of three “seed” polyhedra.

We apply the variational method of Weighted Energy Dissipation (WED) to obtain a global-in-time approximation of solutions to fourth order degenerate parabolic equations of Cahn-Hilliard type. Differently from the standard approach to the existence theory, WED induces an elliptic regularization in time, not in space. The confinement on the phase field is guaranteed already for the approximation, a priori estimates follow without modification of Lyapunov functionals, and the approximations are weakly differentiable in time — even twice in the case of linear mobility. While WED has been widely used for the construction of gradient flows in Hilbert spaces, this appears to be the first application of WED to a metric gradient flow beyond second order PDEs of Wasserstein type.

Moser's worm problem asks for a planar region of minimum area containing a congruent copy of every rectifiable planar arc of length one. We study the Bellman's lost-in-a-forest problem escape path for the isosceles triangle \[ T=\text{conv}\{(-c,0),(c,0),(0,s)\},\qquad s=\frac{766}{\sqrt{625565}},\qquad c=\frac{197}{\sqrt{625565}}. \] The paper gives an exact positive four-source support calibration, a continuum to standard-polygonal reduction, a selected $\Lambda$-gap estimate, a corrected high-angle side-meeting argument, and exact finite ledgers. The original inner-anchor ledger covers 25 temporal orders: its 275 nonzero suffix states reduce to 13 exact squared norms and satisfy $\|R\|<27131/25000$. If either near anchor fails, a new delimiter-gap lemma produces one or two omitted hull edges whose normals range over an exact compact fan interval. A dependency-free rational replay checks 512 one-delimiter orders over 1540 signed half-angle intervals and 4096 two-delimiter orders over a finite signed rectangle cover. These three exhaustive branches give unconditionally \[ E(T)\ge D:=\frac{82074390584}{84861020075} =0.9671624323094728\ldots \] Hence, the convex universal cover of Moser's worm \[ \frac{\text{area}(T)}{D^2} =\frac{11511821678274125}{44639604443512928} =0.257883595112076188\ldots. \]

We show that the $E_8$ and Leech lattices generate the stealthiest isometry-invariant, locally square-integrable point processes of intensity $1$ in $\mathbb{R}^8$ and $\mathbb{R}^{24}$, respectively, and that they are the unique stealthiest processes. We also apply the proof technique to sphere packing and energy minimization, and we construct a $20$-dimensional periodic configuration that is stealthier than any known $20$-dimensional lattice with the same particle density. This periodic configuration is a formal dual of Vardy's $20$-dimensional sphere packing.

This note proves that the truncated octahedron of Archimedes is a strict local minimizer of the isoperimetric quotient among parallelohedra in three dimensions. This extends the corresponding result from the five-dimensional lattice-Voronoi subclass to the full ten-dimensional local parameter space. A six-zone parametrization and Schur's lemma reduce the Hessian to two positive definite $2\times 2$ matrices.

Let $X=\{x_1,\ldots,x_N\}\subset\Sph^{d-1}$ be a spherical $t$-design, $t\ge2$, and let $P_X\subset\R^d$ be its Minkowski polytope. If $h_i=h_{P_X}(x_i)$, then \[ P_X^\circ=\conv\{x_i/h_i:1\le i\le N\}, \] so the design is the radial projection of the polar vertex set. This yields exact degree-dependent moment identities and quantitative control of the unweighted polar moments from the Hausdorff sphericity of $P_X$. Our main results concern self-polarity. If \[ P_X^\circ=cUP_X,\qquad U\in O(d), \] we obtain the structured slack factorization \[ A=c\,hh^T-X^TU^TX,\qquad \rank A=d+1, \] whose zero pattern records the facet–vertex incidences. For node-transitive designs, the common incidence level equals the inradius-to-circumradius ratio $r/R$. Combining this with the one-dimensional moment problem underlying the Fazekas–Levenshtein covering bound, we prove \[ \frac rR\ge\eta_{t,d}, \] with equality forcing the twisted inner-product rows to realize the corresponding Gaussian or Gauss–Radau quadrature rule; in particular, the quantities $N\lambda_k$ become integer incidence multiplicities. We further prove a quantitative near-equality theorem: if \[ \delta=\frac rR-\eta_{t,d} \] is small, then each row is $O_{d,t}(\sqrt\delta)$-close in $W_1$ to the extremal quadrature measure, yielding near-incidence rigidity and an arithmetic stability gap. As applications, we obtain three-dimensional rigidity and identify the regular simplex and the $24$-cell as the regular self-polar examples.

We prove that the regular truncated octahedron uniquely minimizes surface area among all parallelohedra of fixed volume. Equivalently, every three-dimensional parallelohedron $P$ satisfies \[ \frac{\mathcal H^2(\partial P)}{|P|^{2/3}} \ge \frac{3(1+2\sqrt3)}{4^{2/3}}, \] with equality if and only if $P$ is similar to the regular truncated octahedron. Among the non-truncated Fedorov types we prove a stronger sharp bound, attained uniquely by the regular rhombic dodecahedron.

We introduce farthest-cell triplet entropy, the conditional Shannon entropy of the farthest-prototype label given three random prototypes. For independent queries and prototypes, its estimator records only the farthest label, not coordinates or numerical distances. The statistic is bounded by $\log 3$, is invariant under common strictly increasing transformations of the dissimilarities, and has an exact mutual-information interpretation. In high-dimensional isotropic radial models $X_d=R_dU_d$, the Euclidean ordering reduces to scores $\lambda_d\xi_{i,d}-Z_i$, where $\lambda_d=\sqrt d\,\operatorname{sd}(R_d)/\mathbb{E} R_d$ and the $Z_i$ are independent standard Gaussian variables. This gives angular-dominated, intermediate, and radial-dominated entropy limits $\log 3$, $H_{\infty}(\lambda;F)$, and $0$. In hyperbolic space of curvature $-\kappa_d^2$, the same master curve appears at $\lambda_{d,\mathbb H}=\sqrt d\,\tau_d A(s_d)$, where $\tau_d=\operatorname{sd}(R_d)/\mathbb{E} R_d$, $s_d=\kappa_d\mathbb{E} R_d$, and $A(s)=s\coth s$. With a calibrated radial law and $\tau_d$, and a monotone operating interval, entropy inversion identifies the scale-invariant target $s_d^2$; absolute curvature requires an external length unit. CPU simulations give Euclidean and hyperbolic master-curve RMSEs of $0.0164$ and $0.0209$. Inversion from observed synthetic latent coordinates has a median relative error in $\kappa$ of $6.6\%$, while angular anisotropy increases this error to $68.9\%$. Thus the entropy statistic is comparison-based, whereas curvature recovery remains model-calibrated, is not graph-only, and is not robust to anisotropy.

Among four points of a CAT(0) space, the planar symmetric trapezoids are configurations on which Ptolemy's and Reshetnyak's inequalities are both equalities. We show that the symmetric trapezoids remain extremal for the entire family of inequalities interpolating between the two, indexed by the nondecreasing convex functions with concave derivative, and that this function class is characterized by this property. The result is qualitatively stronger than the known quadruple inequalities for this class of functions, and recovers them with their optimal constants, which were previously known only for power functions. Moreover, we extend the analysis to CAT($\kappa$) spaces with $\kappa>0$. We derive variants of Reshetnyak's quadrilateral comparison and of Ptolemy's inequality under positive upper curvature bounds, each with the optimal constant. These two inequalities yield the trapezoid comparison inequality in CAT($\kappa$) spaces, where the product of the bases carries an additional constant factor compared to the $\kappa=0$ case. Again the constant is optimal.

We prove that every $3$-manifold equipped with a proper metric admitting a convex geodesic bicombing has Assouad–Nagata dimension at most $3$. In particular, every proper CAT$(0)$ $3$-manifold has Assouad–Nagata dimension at most $3$. As a corollary, the isoperimetric results of Lang–Stadler–Urech \cite{LSU} and Peteranderl \cite{Pet}, previously applicable to $3$-dimensional Hadamard manifolds, also apply to proper CAT$(0)$ $3$-manifolds of asymptotic rank at most $2$.

We survey biLipschitz (BL) and bounded displacement (BD) equivalence of Delone sets, with an emphasis on examples arising in aperiodic order. After recalling the classical results of Burago-Kleiner, McMullen, and Laczkovich, we discuss criteria for rectifiability, uniform spreadness, and BD equivalence, and how these relate to discrepancy and point-counting estimates. We then focus on Delone sets associated with substitution tilings, where these questions can often be studied through the combinatorial and spectral properties of the underlying substitution rules. We also consider selected classes of cut-and-project sets and briefly discuss extensions beyond BL and BD equivalence.

In this paper we establish removable singularities results for Einstein metrics and for metrics with Ricci curvature bounded below. Let $n\geq 2$. On a closed $n$-manifold, we show that an $L^\infty$-Riemannian metric whose Ricci curvature is bounded below outside a singular set of codimension $> 3- \frac{1}{n-1}$ canonically extends to an $\mathrm{RCD}$ space. As a consequence, using a new removable singularity theorem for Einstein metrics, we prove that in dimension $4$ any Einstein metric with $L^\infty$ singularities of codimension $>3-\frac{1}{3}$ extends smoothly across the singular set, possibly after changing the smooth structure. In higher dimensions, we construct a $C^{1,\alpha}$-Riemannian manifold structure on the regular set of a non-collapsed $\mathrm{RCD}$ space that is a Riemannian manifold with bounded $|\mathrm{Ric}|$ outside a set of codimension $>2$. Our results can be used to give a proof of Schoen's conjecture on scalar curvature singularities for metrics that are either continuous, or $L^\infty$ and sufficiently close to a smooth background metric.

We introduce a framework for establishing Blaschke-Santaló-type inequalities on $m$-tuples of Polish measure spaces coupled together by a continuous cost function. Central to our approach is a transference principle, which provides a mechanism to lift geometric weighted inequalities involving cost-polar sets into functional integral inequalities of Santaló-type. We call these equivalent inequalities cost-Santaló inequalities. This definition expands and includes previous notions in the literature. We apply this principle to deduce several new versions of functional Santaló inequalities, including on the space of rectangular matrices and a functional sine Santaló inequality. A surprising development is that probability spaces with log-concave isoperimetric functions fit into our framework, for example, Gauss space and spherical space, leading to new functional Santaló inequalities in these settings. In particular, we obtain results for $\operatorname{RCD}(K,\infty)$ spaces. As a discrete application, we obtain an inequality for the Hamming cube. Finally, we explore applications to optimal transport, utilizing our functional framework to establish generalized transport-entropy inequalities on arbitrary Polish spaces satisfying a cost-Santaló inequality, which we explicitly instantiate for matrix spaces.

For every $n$, we construct a 2-dimensional $n$-element poset whose Hasse diagram has independence number $n\exp(-\Omega(\sqrt{\log n}))$, and consequently, chromatic number $\exp(\Omega(\sqrt{\log n}))$. This also yields an $n$-point planar set whose box-Delaunay graph satisfies the same bounds.

In this paper, we establish a homological existence criterion for the discrete $L_p$ Minkowski problem for all $p<0$, where $p=-n$ corresponds to the celebrated centro-affine case. For discrete measures with antipodal support, we obtain a complete existence characterization for arbitrary positive masses. In the even case, a solution can be chosen origin-symmetric.

The set of the triangulations with vertex set $X$ of a simple polygon $\mathrm{P}$ can be structured into a flip-graph $\mathcal{F}(\mathrm{P},X)$ whose edges connect two triangulations that differ by a single arc. The geometry of flip-graphs has been thoroughly studied and it is known that the subgraph $\mathcal{F}_\varepsilon(\mathrm{P},X)$ induced by the triangulations that contain a given arc $\varepsilon$ is strongly convex in $\mathcal{F}(\mathrm{P},X)$ when $\mathrm{P}$ is convex and $X$ contains no puncture (points in the interior of $\mathrm{P}$) and at most one flat vertex (points in the interior of an edge). When $X$ contains at least two punctures or flat vertices, it is also known that this strong convexity property fails. Here, we close the last open case by showing that, for any convex polygon with sufficiently many vertices, one can always place a single puncture in $X$ in such a way that $\mathcal{F}_\varepsilon(\mathrm{P},X)$ is not strongly convex in $\mathcal{F}(\mathrm{P},X)$. We prove a similar result for simple polygons with a single reflex vertex. The main ingredients in our proofs are a decomposition lemma for a class of $3$-dimensional triangulations and a hyperbolic volume argument regarding their embedding into $\mathbb{H}^3$.

Posed by Lebesgue in 1914, the universal covering problem asks for the smallest-area planar convex set containing a congruent copy of every set of diameter at most one. We introduce an exact Reuleaux-type variational hierarchy for this constant: its monotone finite-arc values $\Lambda_M$ satisfy $a_{\mathrm{Leb}}=\lim_{M\to\infty}\Lambda_M$, and each level is a continuous finite-dimensional problem. We prove $0\le a_{\mathrm{Leb}}-\Lambda_M\le C M^{-2}$, giving a controlled finite-arc route to the constant itself. As a certified low-order realization, an outward-rounded interval certificate for a regular finite Reuleaux subtest proves $a_{\mathrm{Leb}}\ge0.834$, improving the lower-bound benchmark established by Brass and Sharifi in 2005.

The Modulo Krinkle tilings of Imura (arXiv:2506.07638) form a family of non-periodic, spiral monohedral tilings parametrized by a reduced fraction $m/k$ and an integer $t\ge 2$. We show that the Farey sum (mediant) $(m_1+m_2)/(k_1+k_2)$ of two Farey-adjacent parameters is realized by an exact geometric operation on prototiles: the lower boundary path of the $(m_1+m_2,k_1+k_2)$-prototile is obtained by concatenating the parents' lower paths after an edge-length-preserving progressive rotation (fan-twist) of their edges. Conversely, every prototile admits exactly one fan-twist splice decomposition – no non-adjacent parameters ever splice – the cut position being $k_1=m^{-1}\bmod k$, and the recursion descends the Stern-Brocot tree to a single unit edge. The combinatorial core of the operation is the classical standard factorization of Christoffel words; the contribution here is its exact edge-isometric realization on circular direction systems and the resulting structure theory for the Modulo Krinkle family, including the recently introduced variants: we prove a separation theorem stating that every variant prototile is the common recursively-generated core plus finitely many direction-invariant decoration edges. As a corollary of Imura's spiral-arm count formula, the two Farey parents are visible in the offset-free tiling itself: the numbers of counterclockwise and clockwise spiral arms are $tk_1$ and $tk_2$.

We study a projection identity for a simplex in Euclidean space, written in terms of the frame operator of its unit edge directions. For a right simplex, the identity leads to a sharp family of distance inequalities and a complete description of equality. For a general simplex, the same formula is controlled by the spectrum of the Gram matrix through the Ky Fan principle. We prove converse results that characterise right simplices and determine the smallest number of projection subspaces needed to force orthogonality, together with an optimal quantitative estimate. We also treat affine projection subspaces and show how the original inequality for mutually perpendicular vectors fits into the same framework.

Fix $0<\theta\leqslant 1$. We prove that if $1\leqslant p \leqslant 2/\theta$, then the $\theta$-snowflake of $\ell_2^k$, namely, $\mathbb{R}^k$ equipped with the metric $((x,y)\in \mathbb{R}^k\times \mathbb{R}^k)\mapsto \|x-y\|_2^\theta$, embeds with distortion $O(1)$ into $\ell_p^m$ for some integer $m\lesssim_{p,\theta}k$, which is optimal as $k\to \infty$, as seen by comparing dimensions. However, for $p$ larger than the sharp threshold $2/\theta$ the following change in behavior occurs: If a $(1/\sqrt{k})$-dense subset of the Euclidean sphere $S^{k-1}$ embeds into $\ell_p^m$ with distortion $O(1)$, then necessarily $m\gtrsim_{p,\theta}( k/\log k)^{p\theta/2}$, which grows super-linearly in $k$ as $p\theta/2>1$, and this dimension bound is optimal as $k\to \infty$ up to lower order factors. We deduce from this statement that if $2<p<\infty$, then there exist arbitrarily large $n$-point subsets of $\ell_p$ with the property that if they embed with distortion $O(1)$ into $\ell_p^m$, then necessarily $m\gtrsim_p ((\log n)/(\log\log n)^2)^{p/2}$, thus demonstrating that the statement of the Johnson–Lindenstrauss dimension reduction lemma fails to hold for $\ell_p$

Among three-dimensional parallelohedra of fixed volume, the Archimedean truncated octahedron uniquely minimizes surface area, affirming a conjecture of K. Bezdek from 2006. The final scalar inequality is verified by an exact computer-assisted certificate.

Triangulate a simple quadrilateral by connecting all vertices to an additional point. If the vertices carry values, the piecewise linear function can be assigned a Dirichlet energy. We show that the minimal Dirichlet energy as a function of the location of the inserted point is convex, and the location of the minimum is independent of the values at the corners - a quadrilateral has a discrete harmonic center, characterized by an equilibrium of currents across the inserted edges. It turns out that the fixed points of the Möbius involution swapping opposite corners of the quadrilateral are critical points of this energy, so the discrete harmonic center is Möbius-covariant. For tangential and cyclic quadrilaterals the center admits simple closed forms related to the circle centers. The center and its data-independence generalize to polytopes with d + 2 vertices in dimension d, but the conformal characterizations are special to four points in the plane.

We present a proof classifying continuous isometry invariant valuations on polytopes in hyperbolic space $\mathbb{H}^n$ as linear combinations of the intrinsic volumes. The argument relies on an adaptation of the spherical case recently established by the author that circumvents an approximation step which does not generalize to the hyperbolic setting.

We prove the generalized Geroch conjecture, and the corresponding hyperbolic conjecture, in dimension two.

We develop two methods for proving infinite girth from boundary dynamics. First, we use topologically free extreme boundary actions to give a boundary-dynamical proof of infinite girth for finitely generated acylindrically hyperbolic groups. The second combines Nakamura's criterion with, respectively, flag-space and Roller-boundary dynamics, yielding infinite-girth results for semisimple $S$-algebraic groups and for essential non-elementary actions on finite-dimensional, second countable, non-Euclidean CAT(0) cube complexes. We also prove that every finitely generated large group has infinite girth, thereby answering a question of Akhmedov and Mishra.

Magnitude gives an effective size of a finite metric space and is now used in several data-analysis settings. For such applications, it is natural to ask how magnitude behaves under Gromov-Hausdorff perturbations, including collisions of points. However, magnitude is nowhere continuous on finite metric spaces with the Gromov–Hausdorff topology. We show that this failure is nongeneric in a precise measure-theoretic sense. After fixing the number of points in each collapsing cluster, magnitude is approximately continuous. More strongly, when clusters of points collapse to the points of a limit space, the total weighting of each cluster approximately converges to the weighting of the corresponding limit point. The main tool is a general result on approximate limits of inverses near singular matrices.

We study asymptotic cones of Lie groups, presenting a link between the geometry of the weights and the highly non-simply-connected nature of asymptotic cones. We show that a Lie group that admits a group of SOL-type as a quotient is such that the fundamental group of its asymptotic cone contains the fundamental group of the Hawaiian earring space. We also obtain a strong non-vanishing result for the first homology group of the asymptotic cone. The assumption on the Lie group, introduced by Abels and called the "SOL obstruction", is equivalent to an explicit geometric condition on the weights. Our work builds upon the results of Burillo, who proves the statement on asymptotic cones in the case of SOL, and on those of Cornulier and Tessera who show that the SOL obstruction implies exponential growth of the Dehn function.

We show that if $(X,d,\mathscr H^n)$ is an $RCD(K,N)$ space for some $K\in\mathbb{R}$ and $N\in [1,\infty)$, then it is also a (non-collapsed) $RCD(K,n)$ space. In other words, unweighted finite-dimensional $RCD$ spaces enjoy an automatic improvement of the dimension bound to the Hausdorff dimension of the space. More generally, we prove that this holds also when $N=\infty$, provided that we assume in addition the $n$-rectifiability of the space.

$\newcommand{\X}{\mathbb{X}}\newcommand{\SC}{\mathcal{C}}$ We establish the existence of a "maximally spread out" Borel probability measure on a totally bounded subset $\SC$ of a (quasi)-Banach space $\X$ under two mild conditions: (i) a growth condition on the covering numbers $N(\SC, \epsilon)$ of $\SC$, and (ii) a technical topological condition that is in particular satisfied whenever $\SC\subset\X$ is closed, bounded, and convex. More formally, condition (i) requires that the so-called lower power-exponential Minkowski dimension of $\SC$, i.e., \[s_\ast:=\liminf_{\epsilon\downarrow 0}\frac{\log\log N(\SC,\epsilon)}{\log(1/\epsilon)}\] satisfies $s_\ast>0$. Under these conditions, we construct a Borel probability measure $\mu$ on $\X$ that is critical for $\SC$, or maximally spread out, meaning that the associated outer measure $\mu^\ast$ satisfies $\mu^\ast(\X\setminus\SC)= 0$ and furthermore satisfies for every $0<s<s_\ast$ the small-ball condition \[\mu^\ast(B(x,r))\le\exp\bigl(-c(s)\cdot(1/r)^s\bigr)\quad\text{ for all }x\in\X\text{ and }0<r<r_0(s).\] The existence of such a critical measure in particular implies that the so-called power-exponential Hausdorff dimension of $\SC$ introduced in [J.~Topol.~Anal.~4(2):203–235, 2012] coincides with the lower power-exponential Minkowski dimension. Previous work [Found.~Comput.~Math.~23(1):329–392, 2023] shows that such a critical measure gives rise to a phase transition regarding lossy compression and approximation by quantized neural networks of elements of $\SC$, provided that the $\liminf$ in the definition of $s_\ast$ exists as an actual limit. There, critical measures were constructed for unit balls of certain Besov and Sobolev spaces considered as subsets of $L^2$. In contrast, our construction is completely general. In particular, our results apply to function spaces of dominating mixed smoothness.

A universal cover is a convex set in the plane that contains a congruent copy of every planar set of diameter one. Lebesgue asked in 1914 for one of least area, and the value is not known. We prove that every convex universal cover has area at least 0.8344, improving on 0.832, published in 2005, and 0.833, in a 2026 preprint, both of which come from a disc together with an equilateral triangle and a regular pentagon. Our test sets are instead curves of constant width: the disc, the Reuleaux triangle and the Reuleaux pentagon. Each contains the regular polygon it is built on, so the family is strictly larger at the same number of bodies and the same number of placement parameters, and we show that the classical configuration admits an arrangement whose hull has area below 0.8336, so no bound drawn from those three sets by this argument reaches ours. Curves of constant width were proposed for this role, and explored numerically, by Gibbs in 2014; what is added here is a proof. It consists of an analytic reduction followed by one finite computation. The reduction bounds the hull area from below over an entire box of placements at once, by eroding each Reuleaux polygon to a fixed set contained in every placement that box allows. The computation is an exhaustive subdivision of the resulting five-dimensional space, recorded as a certificate of 486,799,600 nodes and checked by a verifier independent of the search, with a rigorous bound on its floating point error some thousands of times smaller than the margin the verification attains.