Papers

We study the stability of optimal transport for the squared Carnot–Carathéodory distance. We prove the sharp $W_1^{1/4}$ stability estimate for optimal transport maps on complete Riemannian and fat sub-Riemannian manifolds, assuming bounded source support and an upper density bound. On John domains with two-sided density bounds, we obtain logarithmic $L^2$ stability of Kantorovich potentials on complete smooth sub-Riemannian manifolds that are equiregular near the source, with arbitrary compact targets. For potentials, we obtain every Hölder exponent $0<\alpha<1/2$ on two-generating structures, and the exponent $1/2$ in the Riemannian and fat settings and on all step-two Carnot groups. Three-atom examples show that the map and potential exponents with respect to $W_1$ cannot exceed $1/4$ and $1/2$, respectively, on any smooth sub-Riemannian manifold of horizontal rank at least two. We also construct smooth transport families whose target variation of order $t$ in $W_1$ produces map variation of order $t^{1/s}$ in an equiregular region of step $s$. The map estimates follow from coercivity of the duality defect, while the potential estimates follow from quantitative concavity of the Kantorovich functional.

How do algebraic relations among unit directions influence the chromatic number of the plane? We introduce monomial Cayley graphs as a framework for investigating this question, beginning with graphs whose allowed unit steps are signed powers of a single complex parameter. Transcendental parameters give bipartite graphs, while for roots of unity we determine both the chromatic and circular chromatic numbers. As a geometric consequence, every graph admitting a unit-distance embedding in the plane in which the angle between any two edge directions is a rational multiple of $\pi$ is three-colorable. For algebraic noninteger parameters, four colors suffice, and three suffice when the primitive integer minimal polynomial has leading coefficient at least three. We conjecture that three colors suffice for every algebraic noninteger direction parameter and that four suffice for every direction parameter.

A central question in data-driven inverse problems is how to construct a regularizer that adapts to the geometry of the data distribution. Recent work in optimal regularization shows that, within a broad Gibbs class, the regularizer best matched to a distribution $P$ is determined by a single, direction-dependent radial summary statistic $\rho_P$. This suggests a way to control regularizer geometry by transforming the data. In particular, which transformations act on $\rho_P$ in a simple, explicit way? Can they improve the optimization properties of the resulting variational problems or adapt a fixed base regularizer to data? To address these questions, we introduce anisotropic dilation, a direction-preserving map that rescales each point along its Euclidean ray by a positive profile on the sphere. Despite its simplicity, this family acts transitively on the space of radial summary statistics: any target regularizer in the Gibbs class can be reached from any source distribution by a single explicit profile. We characterize the resulting orbit structure on distributions, derive in closed form the profile that optimally adapts a fixed base regularizer to the data, and show that its improvement over isotropic rescaling is governed by a Jensen gap that is provably positive for several natural families. We also establish finite-sample generalization bounds for jointly learning the base regularizer and anisotropic profile, with explicit rates when the logarithm of the profile is parameterized by linear feature models or sparse ReLU networks. Building on this theory, we parameterize the base regularizer and anisotropic profile and learn them jointly from samples, yielding a data-adaptive regularizer for variational inverse problems. Experiments on a controlled two-dimensional family and MNIST denoising show that learning the profile further improves performance.

Let $K=K(n,m,\Gamma)$ be a Bedford-McMullen carpet with $n>m\geq2$ and $1<\#\Gamma<nm$, and let $\mu$ be a self-affine measure on $K$ with strictly positive weights. We prove that if a similitude $f$ satisfies $\mu(K\cap f(K))>0$, then its orthogonal part is diagonal or anti-diagonal. This holds without any arithmetic restrictions on the defining bases. For isometries, we further prove that if $\mu(K\cap f(K))>0$ then $f(K)$ and $K$ differ only by a translation, thereby establishing a strengthened form of the 2010 conjecture of Elekes, Keleti and Máthé in this planar self-affine context.

For every integer $k\ge1$, we prove that each nonempty bounded set in real $\ell_1^{4k+1}$ has an ambient enclosing ball of radius at most $(4k-1)/(4k)$ times its diameter. The bound is sharp in dimension five: $J(\ell_1^5)=3/4$. If a Hadamard matrix of order $4k$ exists, then $J(\ell_1^{4k-1})=J(\ell_1^{4k})=J(\ell_1^{4k+1})=(4k-1)/(4k)$. The upper bound for every $k\ge1$ has been formalized in Lean.

We establish an Alexandrov–Fenchel type inequality in which one of the reference bodies is replaced by a function whose Hessian satisfies a balancing condition on its eigenvalues, without being required to be positive semidefinite. The proof hinges on a new extension of Alexandrov's inequality for mixed discriminants. As applications, we derive a log-concavity principle for linear functionals of area measures, answering a question of Colesanti, Hug, and Saorín-Gómez, as well as Brunn–Minkowski type inequalities for intrinsic volumes of mean section bodies, answering a question of Schuster.

Anisotropic Voronoi diagrams, in which each cell is the region first reached by an ellipse growing from its site, generate non-convex cells whose integral geometry can be described using Minkowski tensors of arbitrary order. In this work, we derive semi-analytical formulas for Minkowski tensors of arbitrary order of the 0th, 1st and 2nd kind for planar anisotropic Voronoi cells that are star convex at their nucleation point. The resulting formulas involve integrals of contact functions, which describe the cell boundary through a diffeomorphic transformation. This transformation renders the geometry amenable to analytical treatment, enabling us to obtain expressions for the desired Minkowski tensors. Although most of the integrals lack closed-form solutions, they can be efficiently approximated numerically once the contact function and its derivatives are known along the boundary of the cell of interest.

A Banach space $X$ is $L$-embedded in its bidual if there is a projection $P\colon X^{**}\to X$ such that $\|x\|=\|Px\|+\|x-Px\|$ for all $x\in X^{**}$. We show that, as long as $X$ has dimension at least $2$, its Lipschitz-free space, denoted by $\mathcal{F}(X)$, is not $L$-embedded in its bidual. Our methods have applications to the problem of when the $L$-embeddability of $\mathcal{F}(M)$ passes to $\mathcal{F}(A)$ for a metric space $M$ and $A\subseteq M$. This is the case when $A$ is compact or when $A$ is geodesically closed.

We study $\varepsilon$-relative approximation of mixed volumes of a fixed number $k$ of full-dimensional convex bodies in $\mathbb{R}^n$, given membership oracles and a known bound $B_n\subseteq K_i\subseteq R_0B_n$. We present a randomized algorithm that estimates any prescribed mixed volume within relative error $\varepsilon$ with probability at least $1-\delta$, using polynomially many oracle calls and bit operations in $n$, $\log R_0$, $\varepsilon^{-1}$, and $\log\delta^{-1}$ for fixed $k$.

Kuperberg conjectured in 1983 that every convex body $K$ in the plane is contained in a quadrilateral of area at most $\frac{3}{\sqrt5}\,|K|$, the extremal bodies being the affine-regular pentagons. On the basis of numerical experiments we propose a stronger conjecture of a purely polygonal nature: every convex polygon $P$ is contained in a quadrilateral of area at most $\frac{3}{\sqrt5}\,|P|$ whose sides are parallel to sides or diagonals of $P$. This chord conjecture implies the inequality conjectured by Kuperberg. For pentagons it is a theorem of Hong, Ismailescu, Kwak and Park, of which we give a short proof by area identities. We also show that the classical bound $\sqrt2$ remains valid for quadrilaterals with sides parallel to chords; the proof is similar to Ismailescu's proof of this bound, but it starts from an inscribed quadrilateral of maximal area instead of a circumscribed quadrilateral of minimal area. We close with a discussion of the difficulties in reaching the constant $\frac{3}{\sqrt5}$.

We prove that finite-dimensional geodesic locally compact Alexandrov spaces with curvature bounded below (CBB), and in particular complete Riemannian manifolds with lower bounded sectional curvature, satisfy a Besicovitch covering theorem by showing that they are directionally limited in the sense of Federer.

In this paper, we investigate the differentiability of Korevaar–Schoen $p$-energy forms and the associated $p$-energy measures. We obtain higher-order derivatives by virtue of explicit realizations of Korevaar–Schoen $p$-energy forms and the associated $p$-energy measures as subsequential pointwise limits of certain double integrals.

The Kannan–Lovász–Simonovits (KLS) conjecture asserts that isotropic log-concave probability measures have Poincaré constants bounded by a universal constant, independently of dimension. We give a deterministic variational proof with an explicit bound on the Poincaré constant $C_P(\mu)\le25$, where $\mu$ is any isotropic log-concave probability measure. Starting from elliptic moment estimates and a quadratic variance inequality, we establish geometric bounds on normalized Appell coefficient norms through joint maximization over the measure and test function. Variation of the measure gives a maximum-principle inequality, while stationarity in the test function controls the highest-order cumulant terms. Two concave barriers constructed from quadratic and cubic polynomials close the induction. A weighted Helmholtz–Hodge decomposition then controls the curl correction of weighted divergence, yielding a curvature estimate for compatible symmetric tensor fields that is uniform in rank. Quadratic duality converts this estimate into an operator comparison for centered integration, linking the coefficient bounds to control of the inverse gradient. A spectral-radius estimate and a scalar growth inequality for adjoint iterates then yield the Poincaré bound. The final estimate is independent of the auxiliary positive curvature, allowing approximation to complete the proof for general isotropic log-concave measures.

Consider the "snaky" corridor $K_4 \subset \mathbb{R}^3$ from Problem 6.64 of arXiv:2511.02864: a tube of unit-square cross-section whose two right-angle turns lie in perpendicular planes, with a straight middle section of clear length $4$. We exhibit explicit rigid motions of connected bodies through $K_4$ and certify their volumes with the problem's own evaluation pipeline, unmodified, at the tight motion tolerances of its final scoring stage (translation $0.005$, rotation $0.05$ degrees). The certificates give $V^*(K_4) \ge 1.8355162$ at grid fineness $0.002$ and $V^*(K_4) \ge 1.8558404$ at grid fineness $0.001$, against the previously reported $1.81$; rerunning the pipeline's published best path under the identical configuration yields $1.8179428$, so every comparison is path against path under one fixed certifier. Three stages produce the record motion: a simple "staggered twist" backbone in which the whole 90-degree twist about the tube axis happens inside the second turn; evolutionary (CMA-ES) refinement, including a mixed-fidelity scheme that re-scores each generation's leading candidates on a grid twice as fine and lets those ranks steer recombination; and a gradient polish carried out directly at the finer carve resolution, guarded by periodic rebuilding of the frozen interpolation structure and by a penalty on control-pose spacing. These guards close three documented failure modes in which an optimizer inflates its own objective: frozen-structure inflation, spacing stretch, and exploitation of a coarse grid by optimizing and selecting on it. A staircase-based predictor with two grid bases, anchored by matching grid sensitivity, forecast the certificates to about $0.001$ before the runs. With the companion note's upper bound $V^*(K_4) \le 3.4083$, the problem now sits in the bracket $1.8558404 \le V^*(K_4) \le 3.4083$. All paths, notebooks and certified results are public.

Let $K_M \subset \mathbb{R}^3$ be the "snaky" corridor with unit-square cross-section: an infinite entry arm parallel to the $x$-axis, a middle arm parallel to the $y$-axis whose clear straight section has length $M$, and an infinite exit arm parallel to the $z$-axis, with the two turns lying in the $x$-$y$ and $y$-$z$ planes. Let $V^*(K_M)$ be the supremum of the volumes of connected bodies that can be moved continuously (by rigid motions) from the entry arm to the exit arm while remaining inside $K_M$. We prove $V^*(K_M) \le 7.8691$ for every $M \ge 3.53$ (Theorem A); by a sharper analysis of the same mechanism, $V^*(K_M) \le 4.2862$ for every $M \ge 2.61$ (Theorem B); and, by replacing the final analytic caps with a certified branch-and-bound computation over the compact family of entry tubes compatible with the forced events, $V^*(K_M) \le 3.4083$ for every $M > 2.2$ (Theorem C; computer-assisted, in double precision with conservative slack). In particular all three bounds apply to the corridor with $M = 4$ used in Problem 6.64 of Georgiev, Gómez-Serrano, Tao and Wagner (arXiv:2511.02864), and uniformly to every "turns far apart" reading of that problem. To the best of our knowledge, this is the first upper bound for the snaky-corridor 3D moving sofa problem; previously not even finiteness of $V^*$ had been recorded. This note proves upper bounds only; combined with the certified lower bound 1.8558404 from the author's separate certification work via the problem's own evaluation pipeline, the state of the problem becomes $1.8558404 \le V^*(K_4) \le 3.4083$. Theorems A and B are elementary and self-contained: a first-crossing (elbow) event forces a definite tilt of the body's own entry tube, and Fubini-type chord estimates for tilted tubes cap the volume. Theorem C certifies, by the same forced-event skeleton, the largest corridor volume that any admissible entry tube can carry.

Let $\varepsilon_1,\ldots,\varepsilon_n$ be independent Rademacher random variables. We settle the best constant $c$ such that \[ \mathbb{E}\Bigl|\sum_{j=1}^n a_j\varepsilon_j\Bigr| \ge \frac{1}{\sqrt{2}} +c\Bigl|a-\frac{e_1+e_2}{\sqrt{2}}\Bigr| \] holds for every $n\ge 2$ and every unit vector $a\in\mathbb{R}^n$ with $a_1\ge\cdots\ge a_n\ge 0$. We prove that the optimal constant is $c=(2-\sqrt{2})^{3/2}/8$, with equality attained at $a=\frac{1}{2}(1,1,1,1,0,\ldots,0)$ for $n\ge 4$. This completes a line of research by De-Diakonikolas-Servedio, Eskenazis-Nayar-Tkocz and Fang-Wang.

We study a Poincaré-type inequality for Gaussian measures restricted to origin-symmetric convex sets. The inequality arises from the second variation at $t=0$ of the function $t\longmapsto \gamma_{\Sigma}\left(e^{tA}K\right)$, where $\Sigma$ and $A$ are symmetric matrices. The relevance of this function comes from Saroglou's reformulation of the log-Brunn–Minkowski conjecture. We show that $\Sigma$ and $A$ commute if and only if the inequality holds for every box of the form $UD B_\infty^n$, where $U$ is orthogonal and $D$ is diagonal. In the non-commuting case, this yields counterexamples for orthogonal boxes whose axes can be chosen arbitrarily close to the coordinate axes. We also show that the inequality holds when $K$ is unconditional with respect to an eigenbasis of $\Sigma$ and $A$ is an arbitrary symmetric matrix.

We settle the long-standing Godbersen conjecture that the mixed volume of a convex body of fixed positive volume with its reflection about the origin is maximized precisely by simplices. For the proof, we establish a new characterization of simplices by a natural incidence relation on the face lattice. Moreover, we establish a higher-order generalization of the Godbersen inequality and show that simplices are the only nontrivial equality cases, confirming a conjecture of Schneider from 2000.

In this series of articles, we study two graph classes and their various geometric representations. The first class consists of strongly involutive self-dual graphs (SISD graphs); the second consists of non-bipartite quadrangulations of the projective plane (NBQP graphs). These classes naturally arise in many geometric problems. For example, NBQP graphs are connected to diameter graphs in $\mathbb{R}^3$, tangency graphs of families of pairwise intersecting circles and pseudocircles, generalized thrackles, and graphs formed by the main diagonals of negatively self-polar polytopes. On the other hand, SISD graphs correspond to the skeletons of negatively self-polar polytopes, self-dual cones, and extremal ball-polytopes. In Part I we show a natural one-to-one correspondence between non-bipartite quadrangulations of the projective plane and 2-connected strongly involutive self-dual maps. In particular, we show that the reduced vertex-face incidence structure of a 2-connected strongly involutive self-dual map is a non-bipartite quadrangulation of the projective plane, and every non-bipartite quadrangulation of the projective plane arises that way. This allows us to translate natural properties from one class to the other. These results lay the foundation for later parts of the series.

A unit-distance graph is a graph which admits a realisation in Euclidean space in which every edge has unit length. Imposing further geometric conditions on the non-edges gives a family of natural subclasses. Requiring that no two vertices lie at distance less than one gives penny and marble graphs: the contact graphs of collections of equal radii $d$-dimensional spheres with non-overlapping interiors for $d=2,3$. Requiring instead that the straight-line drawing in the plane be non-crossing gives matchstick graphs. Since any motion of a realisation must preserve these extra conditions, the rigidity and flexibility properties of the resulting frameworks differ from those of classical bar-joint rigidity theory. In this note we analyse various rigidity problems for penny and marble graphs, matchstick graphs and unit-distance graphs. In particular we answer a recent open problem on penny and marble graph rigidity and establish a link between penny graphs and the concept of NAC-colourings.

The minimum solid angle of a simplex is shown to be a shape measure in any dimension. While this has been known in dimensions two and three, the higher-dimensional generalization is new. We prove inequalities between the minimum solid angle of a non-degenerate simplex and its fullness. Up to dimension-dependent constants, the fullness is bounded from above by the minimum solid angle and from below by the square of the minimum solid angle. The exponents are optimal for three and higher dimensions. This work therefore generalizes the minimum angle condition. We also review several simplex shape measures and their relationships.

We introduce temporal metric spaces, a framework for formalizing metric and topological structures on continuously evolving domains. Rather than defining a single metric on a fixed space, we construct a family of interval-dependent metrics on an associated trajectory space, enabling distances to be measured between continuous trajectories over finite time intervals. We establish the fundamental topology of the trajectory space, prove that it inherits completeness from the underlying metric space, and develop the corresponding temporal hyperspace together with its completeness. Finally, we develop a theory of iterated function systems acting on these evolving spaces, proving that their attractors evolve coherently with the underlying dynamics and remain stable even when the evolving geometry causes the generating mappings to lose their classical contraction property.

Let $Q_n=[0,1]^n$, $n\ge2$. Denote by $M_n$ be the maximal volume of a pyramid in $Q_n$ whose base is a hyperplane section of $Q_n$, and by $M_n^0$ the corresponding maximum when the section contains the center of $Q_n$. We prove that $nM_n<(n+1)M_{n+1}^0$.

Together with Bourgain's slicing problem and the variance conjecture, the Kannan-Lovász-Simonovits (KLS) conjecture has been a driving force for mathematical developments in high dimensional geometry in recent decades. We present here a proof of this conjecture. The crucial step in the proof is the very recent criterion from Song and Zhang involving high-order derivatives of tilt averages.

For each fixed $n$, we prove a positive lower bound for the diameter of a compact simple geodesic $n$-gonal monotile of the hyperbolic plane. We also prove a positive lower bound for the area of such a tile in a finite monohedral tiling of a closed hyperbolic surface, independent of the topology and metric. Both bounds become independent of $n$ for vertex-proper tilings, in which every genuine tile vertex belongs to at least three distinct tiles. This includes tilings by convex polygons. Reflex angles and non-edge-to-edge incidences are allowed. After qualitative proofs, we obtain explicit constants from a sharp gap estimate for packing polytopes with arbitrarily coupled nonnegative integer constraints. Exact corner balance applies on closed surfaces; in the plane, covering duality and ball counts give a boundary factor depending on the tile diameter. Following Zare, we give, for each integer $q\ge2$, a geodesic construction with $2q+3$ sides and diameter less than $3/q$, showing that side counts cannot be unrestricted without an additional condition. The closed-surface results extend to regular curved sides. The geometric constants are effective but are not claimed to be optimal.

We study three affine variational quantities built on the affine $L^p$ energy of Lutwak, Yang and Zhang: the affine Poincaré–Sobolev constants $\Lambda^{\mathcal{A}}_{p,q}(\Omega)$ of a bounded open set, the associated affine torsional rigidity, and the affine $p$-capacity of a compact set. We show that ellipsoids are the only extremal sets in their corresponding isoperimetric inequalities, and that the affine capacity of a compact set of positive volume is attained by a potential solving the affine $p$-Laplace equation. We also prove an affine Kohler-Jobin inequality: among sets with prescribed affine torsional rigidity, only ellipsoids minimize $\Lambda^{\mathcal{A}}_{p,q}$. Following the approach of Haddad, Jiménez and Montenegro to affine Sobolev inequalities, our methods rest on the $L_p$ Busemann–Petty centroid inequality. Proving a stability version of it, we obtain quantitative forms of the isoperimetric inequalities for $\Lambda^{\mathcal{A}}_{p,q}$, the affine torsional rigidity and the affine $p$-capacity, together with a stability result for the affine Kohler-Jobin inequality on convex sets.

We construct smooth collapsing sequences with sectional curvature at least $-1$ whose limits are topological manifolds with nonzero Kirby–Siebenmann class. For every $n\geq6$, the source and limit are simply connected, have dimensions $n+3$ and $n$, and the metrics have a common positive Ricci lower bound. The limits are homeomorphic to the product of the Chern four-manifold with $S^{n-4}$. We also obtain spin examples and five-dimensional lens-space homotopy types, the latter from finite quotients of one fixed $2$-connected eight-manifold. Finally, we realize aspherical manifolds with no closed PL representatives of their homotopy types as collapsing limits with scalar curvatures tending uniformly to $+\infty$, and establish an obstruction to their approximation with almost nonnegative Ricci curvature.

It is well-known that a random Steiner process converges almost surely to a Euclidean ball. On the other hand, as we show in this article, a planar random shaking process converges almost surely to a triangle (in the Banach-Mazur distance). Asking whether these phenomena are generic or exceptional, we study random iterations of an operation called $\lambda$-Steiner symmetrization, which includes Steiner symmetrization and shaking as special cases. We establish abstract criteria for almost sure convergence and divergence of such processes, obtaining criteria for many other symmetrization operations from convex geometry as a byproduct. Based on a joint characterization of ellipses and triangles, we give a full resolution to our question in the planar case.

We introduce a quantum coding framework for discrete small-shift noise, in which errors on qudits are modeled as low-weight $X$- and $Z$-type Pauli shifts, analogous to small phase-space displacements in continuous-variable systems. This structure is approximately respected by nuclear-spin noise and captured by the Lee metric, motivating a quantum extension of classical Lee-metric coding theory. We develop such a formalism for stabilizer codes over $\mathbb{Z}_q$ for arbitrary $q$, using joint and separate Lee metrics for the $X$- and $Z$-components of errors. For qubits, the joint metric counts $Y$ errors twice and can yield codes that detect and correct $X$ and $Z$ errors with fewer physical qubits than codes designed for the conventional Hamming metric. We discuss Lee-weight spreading under Clifford gates and qubitize codes over $\mathbb{Z}_4$ via the Gray map, finding codes with two-fold transversal non-qubit-Clifford gates. For quantum CSS Lee-LDPC codes on $n$ qudits, we prove that the Lee distance cannot exceed $O(n)$, uniformly in $q$, demonstrating an unexpected obstruction to using the large internal Hilbert space of a large-$q$ qudit to make high-Lee-distance codes. Under local Metropolis dynamics, certain classical $q$-ary “helical repetition codes” have exponentially long memory times at fixed temperature for sufficiently large $q$ (that grows with system length), which can be understood as spontaneous symmetry breaking at finite temperature, even in local one-dimensional models. Hypergraph products of these helical repetition codes provide local two-dimensional quantum codes that inherit self-correction for $Z$ errors, but self-correction for $X$ errors remains an open question.

We provide a necessary condition for integral polytopes to satisfy subspace concentration type inequalities, by mean of a convex geometric interpretation of the Bogomolov–Gieseker inequality.