Papers

Godbersen's long-standing conjecture states that the mixed volumes of any convex body $K \subset \mathbb R^n$ satisfy $V(K[k], -K[n-k])\le \binom{n}{k} \operatorname{vol}(K)$ for any $k \in \{1, \dots, n - 1\}$, with equality if and only if $K$ is an $n$-simplex. Kotrbatý and Mouamine recently proved the inequality in full generality, and showed that if $K$ is a (full-dimensional) polytope attaining equality, it must be a simplex. We complete the characterization of the equality case: if $K \subset \mathbb R^n$ is a convex body for which $V(K[k], -K[n-k]) = \binom{n}{k} \operatorname{vol}(K)$ for some $k \in \{1, \dots, n -1\}$, then $K$ is an $n$-simplex. By a similar method, we characterize the equality case in the higher-order Godbersen inequality conjectured by Schneider and by Kotrbatý.

Let $P\subset\mathbb R^d$ be finite with $\operatorname{diam}(P)\le1$. For $0<\varepsilon<1$, let $\mathcal N_\varepsilon(P)$ denote the number of ordered pairs at distance at most $\varepsilon$, including the diagonal, and let $\mathcal A_\varepsilon(P)$ denote the number of ordered pairs at distance at least $1-\varepsilon$. We prove the sharp bound \[ \mathcal N_\varepsilon(P)\gtrsim_d \varepsilon^{(d-1)/2}\mathcal A_\varepsilon(P) \] conjectured by Steinerberger (2025). More generally, the same inequality holds for finite positive Borel measures. The main input is a uniform $L^2$ estimate, at cap thickness comparable to lattice spacing, for a supporting-direction incidence operator. In gnomonic coordinates, this operator is controlled by an incidence estimate for small Fenchel–Young gaps.

We develop a theory of orientability for metric manifolds and singular metric spaces using locally integral currents. For topological $n$-manifolds with locally finite Hausdorff measure, we introduce local degree and measure-theoretic conditions under which topological orientations correspond canonically to boundaryless, locally integer rectifiable $n$-currents of multiplicity one on the rectifiable part. We extend these currents to a class of singular spaces whose manifold part is dense and of full measure, and establish constancy and top-dimensional homology results under local uniqueness of orientation. For orientable non-collapsed $\RCD(\kappa,n)$ spaces, we prove a current-theoretic Stokes theorem: the boundary of the orientation current is the induced orientation current of the geometric boundary, with multiplicity one. In particular, its mass measure is precisely the $(n-1)$-dimensional Hausdorff measure on that boundary. For purely $n$-dimensional, locally geodesically complete spaces with local upper curvature bounds, we show that local uniqueness of orientation excludes codimension-one branching and implies a local $1$-Poincaré inequality. The latter result applies beyond homology manifolds and extends the connection between quantitative topology and Poincaré inequalities to this class of singular spaces.

In this paper we introduce a geometrically constrained optimal transport problem on Riemannian manifolds endowed with a Borel foliation, in which mass is transported only within individual leaves. We characterize the existence of foliated transport plans through the equality of the transverse projections of the marginals and show that the corresponding quadratic transport cost decomposes into a measurable family of leafwise optimal transport problems. We also formulate the problem as a classical Kantorovich problem with an extended-valued leafwise cost. Finally, we introduce a foliated continuity equation and establish a Benamou-Brenier formula relating the static and dynamical foliated formulations.

We present several geometric constructions of graphs with rapidly growing chromatic numbers, most of which are triangle-free or have large girth.

We give two alternate formulations for the second moment formula of the Siegel transform over $\mathrm{Sp}(2n, \mathbb{Z}) \backslash \mathrm{Sp}(2n,\mathbb{R})$, originally established by Kelmer and Yu. We also provide several applications that demonstrate their usefulness.

We prove that metric measure spaces arising from $2$-manifolds equipped with Riemannian metrics $g$ such that $g, g^{-1} \in L^\infty_{\rm loc} \cap W^{1,p}_{\rm loc}$ are infinitesimally Hilbertian. Along the way, we establish uniform lower $1$-capacity bounds of several $1$-dimensional subsets of $\R^2$.

We derive the sub-Riemannian geodesics of the affine-additive group using the Pontryagin Maximum Principle (PMP) and compute the associated cut times. We then proceed to construct the Carnot-Carathéodory sphere of the affine-additive group as well as the flask surface; the latter is the analogue of the Pansu sphere (bubble set) in the first Heisenberg group.

We present a simple proof of the $\varepsilon$-Dvoretzky conjecture, which asserts that the dependence on the approximation parameter $\varepsilon$ in Dvoretzky's theorem is polynomial in $1/\varepsilon$. In particular, if $n \geq (C/\varepsilon)^{\ell/2+1}$, then any $n$-dimensional convex body has, through any given interior point, an $\ell$-dimensional section that is $\varepsilon$-close to a Euclidean ball. Here, $C > 0$ is a universal constant. We in fact obtain a sharper dependence on $\varepsilon$. The proof is probabilistic, but uses a different probabilistic model from those employed previously. We also prove a simultaneous version of our theorem for a finite family of convex bodies with the origin in their interior, yielding a common linear subspace on which all of the bodies have nearly-Euclidean sections. Finally, we compare the radius of the approximating Euclidean ball with familiar geometric parameters, such as the mean widths of the body and its dual.

We characterize lower curvature bounds on metric surfaces by the Bishop–Gromov inequality for metric balls.

Problem 73 on Green's list of open problems asks whether a smooth submanifold of codimension two in $\mathbb{R}^n$ must meet some affine two-plane in five points when $n$ is large. In this paper, we answer this question for every $n \geq 6$. Every nonempty relatively open subset of such a submanifold contains the vertices of an affinely regular pentagon, and the bound $n \geq 6$ is sharp for affinely regular pentagons. More generally, for every prime $p \geq 5$, the graph of a continuous map from a ball in $\mathbb{R}^{p-1}$ to $\mathbb{R}^2$ contains an affinely regular $p$-gon. A second argument gives six coplanar points when $n \geq 7$. Both proofs find the points by making the low Fourier modes of the graph map vanish along a circle.

We present a version of the vector balancing problem in which each vector may be given a sign and a permutation of its coordinates. We prove that this vector balancing problem and its corresponding prefix problem admit an explicit bound, and we further show that it is asymptotically optimal in the dimension. Our method of proof is purely geometric.

A net of conics can remain unchanged as the four-point configuration generating it varies. We study this phenomenon over a field ${\mathbb{k}}$ of characteristic different from $2$. We allow degenerate quadrangles, consisting of four not necessarily distinct points and six joining lines, provided that three sides form a nondegenerate triangle. The net of a quadrangle is obtained by adjoining constants to the defining polynomials in the pencil generated by its pairs of opposite sides, viewed as conics. We call two quadrangles entangled when they generate the same net. We prove that entanglement is equivalent to the apparently asymmetric condition that every side of one quadrangle bisects the other quadrangle at that side's midpoint. The line-pair members of the net form a bisector field that determines the entire net, so entanglement is also equivalent to equality of bisector fields. Finally, we prove that the locus of centers of the central conics in the net, together with one nonparallel line-pair member, determines the net.

In [H. Lavenant, J. Luckhardt, G. Mordant, B. Schmitzer, L. Tamanini, The Riemannian geometry of Sinkhorn divergences. Ann. Inst. H. Poincaré Anal. Non Linéaire 43 (2026)] we introduced a Riemannian metric $\mathsf{d}_S$ on the space of probability distributions obtained from entropic optimal transport, specifically from the Sinkhorn divergence $S_\varepsilon$. In the present work we discuss how to approximate and compute $\mathsf{d}_S$. Spatially, we prove Gromov–Hausdorff convergence of the metric and convergence of geodesics for increasingly fine Eulerian discretization of the base space. Temporally, we show $\Gamma$-convergence of the chain discretization $N \sum_{k=0}^{N-1} S_\varepsilon(\mu_k, \mu_{k+1})$ to the energy functional defining $\mathsf{d}_S$. We deduce and implement numerical schemes to compute approximations of $\mathsf{d}_S$.

We develop a measure-theoretic framework for small-order limits of nonlocal energies with metric-valued maps. For globally $L^p$ maps, we consider interaction measures whose marginals have uniformly bounded $L^\infty$ densities $a_\alpha$ and $b_\alpha$. If these densities converge weakly-* to $a_\infty$ and $b_\infty$, the normalized energies converge to the $L^p$ energy weighted by $a_\infty+b_\infty$ if and only if the interaction measures escape every bounded rectangle. For maps that are only locally $L^p$, we introduce target laws describing the distribution of the values sampled at infinity. Convergence of their $p$-distance profiles yields the limiting interaction energy for every $1\leq p<\infty$ and arbitrary Polish targets, with integrated $p$-Wasserstein convergence as a sufficient criterion. The arguments include the endpoint $p=1$ and require no linear structure on the target space. Applications include directional and anisotropic Maz'ya–Shaposhnikova formulas, a Bernoulli-law interpretation of fractional perimeters, and an Abelian principle for heat-semigroup energies.

Let $K$ be any complete non-Archimedean field, not necessarily spherically complete. We prove that every polystable representation of a finite quiver over $K$ admits a split harmonic norm in the sense of Haiden–Katzarkov–Kontsevich–Pandit (HKKP), answering a question from their preprint Towards Categorical Kähler Geometry. Moret-Bailly's closed-image theorem gives a uniform separation from representations with destabilizing subobjects. A finite estimate on norm weights then yields a linear coercivity bound for stable representations and hence a minimum of the extended HKKP potential on the complete CAT(0) space of all norms. A direct first-variation computation identifies split minimizers with HKKP harmonic norms. Strict supporting hinges reduce split attainment to semistable reduction for quivers with arbitrary real radius labels. Small changes of the lifted splitting make each reversed extension nonsplit; a finite slope–rank descent proves termination. Consequently, every minimum in the completed norm space is attained by a split norm with the same arrow singular profiles, without restrictions on the value group or the quiver. Nonsplit minimizers illustrate split attainment in one and two radius modifications, and a two-parameter Berkovich analytic family separates polystability, semistability, and harmonicity of a chosen split norm.

The honeycomb and Kelvin problems ask how space can be partitioned into cells of equal volume with the least surface area. We resolve this question in four dimensions for convex bodies that tile space by translations, showing that the regular $24$-cell uniquely minimizes surface area at fixed volume. We develop a framework combining an exact deficit identity, valid in every dimension, with a descent argument. As a consequence, the regular $24$-cell also minimizes surface area among such tiles containing a fixed ball.

Following Miyamoto, we consider Q-charts, a class of subductions $\phi\colon X\to Y$ of diffeological spaces along which Blohmann's elasticity descends. We show that the Riemannian apparatus of elastic diffeology—vertical, horizontal and full connections, Riemannian metrics, Levi-Civita connections and geodesics—also descends along Q-charts. More precisely, structures on $Y$ correspond bijectively to structures on $X$ that are invariant under the pseudogroup $\Psi(\phi)$ of fibre-preserving transitions, and torsion-freeness, effectivity, flatness, fullness, the Levi-Civita property and the geodesic equation are preserved in both directions. The key input is that the iterated tangent bundles $T^nX$ and their fibre products $T_kX$ are the pullbacks along $\phi$ of the corresponding bundles over $Y$. For quotients by principal actions of diffeologically discrete groups, we lift geodesics globally, prove that the class of elastic spaces on which $\R$ is a curve object is closed under such quotients, and show that complete geodesic flows descend. As an application, we determine the Riemannian geometry of the irrational tori $\Tor_\alpha=\R/(\Z+\alpha\Z)$ completely. Let $\vartheta$ be the one-form on $\Tor_\alpha$ induced by $dt$ on $\R$. Every Riemannian metric is a positive constant multiple of $\vartheta^2$, and the vertical connections form a one-parameter family of flat effective connections, a single member of which is the unique Levi-Civita connection of every metric. This connection has a complete geodesic flow, yet the induced pseudo-distance vanishes identically.

Trisecting the dihedral angles of an $n$-simplex defines its Morley simplex. We study the original simplices for which this simplex is regular. A criterion in terms of the Gram matrix of the facet normals reduces the problem to a matrix equation. The derivative of the Morley map at the regular simplex has two explicit eigenvalues, both nonzero for $n\ge3$; thus the regular simplex is an isolated solution. We conjecture that in dimensions four and five a regular Morley simplex forces two hyperplane reflections interchanging disjoint pairs of vertices. We prove that this conclusion fails in every dimension $6\le n\le200$ and in every dimension $n=\binom k2-1$ with $k\ge9$: in these dimensions there are simplices with regular Morley simplex and no hyperplane reflection symmetry. The examples for $8\le n\le200$ have dihedral symmetry of order $2(n+1)$, while the infinite family is based on the Johnson scheme. In dimension four we give exact constructions of two nonregular examples, defined by irreducible polynomials of degrees $18$ and $8$ with Galois groups $S_{18}$ and $S_8$. Neither example is expressible by radicals. The computer assisted existence proofs use exact rational arithmetic and intervals with outward rounding.

We characterize admissible BLD remetrizations of branched self-covers of spheres by linear local contractibility of their canonical pullback length metrics. A relative homology obstruction involving the total degree of an inverse image component yields uniform porosity of branch values. Explicit degree-two cusp covers of $\mathbb S^n$, $n\ge3$, have uniformly porous branch values of Hausdorff dimension $n-2$ but admit no compatible Ahlfors $n$-regular, linearly locally contractible BLD source metric. Short homologically essential loops provide the obstruction, showing that porosity is necessary but insufficient for admissibility. We also prove quasisymmetric invariance and sharp degree dependence of porosity for sphere-to-sphere covers in dimension two.

The discovery of Chair44 (Tsiokos, 2026) settled the three-dimensional einstein problem with a strongly aperiodic polyhedral monotile in $\mathbb{R}^3$. This note extends the underlying mechanism—the rep-$2^N$ chair $C_N = [0,2]^N \setminus (1,2]^N$ with corner/socket markings—to $\mathbb{R}^N$. Besides expository material (the rep-$2^N$ dissection and a conditional strong-aperiodicity theorem under lattice registration and hierarchical enforcement), the note makes a new computational contribution. We introduce a frame-marking formalism in which the marking of a tile is its full orientation frame and the matching rule is the contact language generated by the substitution itself; this makes the search for matching rules finite in every dimension. We give a finite certificate (coarsening closure, tightness, and a two-shell enclosure analysis) whose validity implies that every lattice-registered tiling by the marked tile is uniquely hierarchical, hence strongly aperiodic. For $N=3$ the certificate passes: it yields explicit facet matching rules on the 24 panels of $C_3$ (135 admissible facet-contact triples) and reproduces, from first principles and independently of published constructions, the Chair44 statistics 2388 $\to$ 44 admissible contacts (30 occurring), 33 one-shell clusters, 15 extendable, each forcing a unique supertile. Among the 2187 homochiral frame assignments of the 3D substitution with a translated central child, the certified one is unique up to conjugation. For $N=4$ the same pipeline is run on several structured families of frame assignments (canonical, $D_4$-, $Z_2\times Z_2$- and $Z_4$-symmetric, and a lift of the 3D solution); none is coarsening-closed, and we report the failure data. A self-similar marking of $C_4$ thus remains an explicitly finite, open computational problem, which we state precisely. Code: https://github.com/dimkadimon/Monotile-RN

Let \(N_t\) denote the least number of nodes in a positive cubature formula of degree \(t\) on \(S^2\). We prove that a formula of degree \(2m+1\) cannot have exactly \((m+1)(m+2)+1\) nodes for any \(m\ge2\). Excluding equality in the Fisher bound then gives \[ N_{2m+1}\ge (m+1)(m+2)+2 \] for \(m\ge3\), and known constructions yield the exact values \(N_7=22\) and \(N_9=32\). For general odd degrees, positive circle measures on Lobatto latitudes give an upper bound with quadratic coefficient \(13/8\), parity-dependent linear terms, and an \(O(m^{2/3})\) remainder. We determine the sharp constant \(49\sqrt[3]{3}/72\) for the scalar remainder in this construction. Lower bounds are obtained from continuous weighted LP–Turán inequalities and radial caps whose Helmholtz companions are nonnegative measures. We prove that the cap functional admits a maximizer at each fixed admissible support radius and derive explicit finite-degree lower bounds by a positivity-preserving transfer to the sphere.

We determine the exact number, up to rotations, of nonconstant positive $C^2$ solutions to the planar isotropic $L_p$ dual Minkowski problem for every $(p, q) \in \mathbb{R}^2$. We obtain a new parametrization of the associated period integral, characterize the regions where the period is strictly increasing or strictly decreasing, and prove that in the remaining nonmonotone region it has a unique nondegenerate maximum. As a consequence, we have a complete classification.

In this article, we investigate the Assouad spectrum and Assouad dimension of graphs of fractal functions generated by generalized affine iterated function systems. We establish upper and lower bounds for the Assouad spectrum in terms of the scaling functions and the underlying partition of the generalized affine construction. If the scaling function is Lipschitz continuous and the partition is uniform, we obtain an explicit expression for the Assouad spectrum of the associated graph. These results provide a connection between the parameters defining the generalized affine fractal function and the local multiscale geometry of its graph. As applications, we consider two classic examples of generalized affine fractal functions, namely the Weierstrass and Takagi functions. For the classical Weierstrass function $W$, whose graph $\Gamma_W$ has the box dimension $2+\log_N\lambda$, we obtain \[ \dim_A^\theta(\Gamma_W) \leq \frac{2+\log_N\lambda-\theta}{1-\theta}, \qquad \theta\in \left(0,\log_N\frac{1}{\lambda}\right). \] and if $\lambda^2 N <1$, we get \[ \dim_A(\Gamma_W)\geq 1 +\log_N\left(\frac{1}{\lambda}\right). \] For the classical Takagi function $T$ with graph $\Gamma_T$, we show that \[ \dim_A^\theta(\Gamma_T)=1, \theta\in(0,1), \] and consequently its quasi-Assouad dimension is equal to $1.$ These results settle an open problem on dimension of graphs posed by Fraser.

We give a full list of translational nets which flex within their class of discrete surfaces of translation, by reducing the classification problem to the one of flexible complete bipartite frameworks on the sphere, for which the solution is known. We also obtained two novel classes which correspond to Bottema's spherical 16-bar mechanisms and the constant diagonal angle frameworks. Based on an algorithm for the construction of all flexible translational nets, we also discuss flexible translational tubes and toroids. Furthermore, we present novel results for both topologies which are implied by Bottema's spherical 16-bar mechanisms.

This survey presents heat kernel regularization as a method for quantitative stability of Kantorovich potentials for the quadratic transport cost. We give a complete new heat kernel proof on Heisenberg groups equipped with the Carnot–Carathéodory distance. For source densities bounded above and away from zero on bounded John domains, we obtain $L^2$ stability with rate $W_1^{1/2}$ for arbitrary targets in a fixed compact set, including atomic measures. We also establish a new $L^2$ stability estimate on finite-dimensional RCD spaces, with the optimal dimension-independent exponent $1/2$.

Radchenko and Wheeler (RW) recently proved a finite pentagon relation for real quadratic special values of the modular quantum dilogarithm and used it to establish the rank-$1$ twisted convolution identity conjectured by the current authors. RW gave an explicit argument in the principal case and remarked that their proof holds for all rank-$1$ admissible tuples. We extend their proof to all rank-$r$ admissible tuples and provide an explicit dictionary between the modular quantum dilogarithm and the Shintani-Faddeev modular cocycle conventions in the respective papers. Thus, we establish that, if $d,r$ are positive integers such that $r<\frac{d-1}{2}$ and $\frac{d^2-1}{r(d-r)} \in \mathbb{Z}$, then there exist ghost $r$-SICs: i.e., configurations of $d^2$ rank-$r$ subspaces in $\mathbb{C}^d$ that satisfy a non-Hermitian equichordal condition. Under the Stark conjecture, these configurations are Galois conjugate to Hermitian equichordal configurations called $r$-SICs (or rank-$r$ SIC-POVMs).

In this paper, we consider certain uniformity conditions on the radii of balls. Under these conditions, we study the semicontinuity of the Hausdorff dimension of measures with respect to measured Gromov–Hausdorff convergence and box convergence. We also examine the semicontinuity of other dimensions of measures, which are similar to the Hausdorff dimension of measures.

We show that, given a partially defined metric $d:R\rightarrow\mathbb{R}$ where $R\subset X^2$ is a closed equivalence relation on a compact metrizable space $X$, there is a metric $\hat{d}:X^2\rightarrow \mathbb{R}$ such that $d = \hat{d}|_{R}$

These notes introduce the martingale dimension of a strongly local regular Dirichlet form and the analytic dimension of a Lipschitz differentiability space. We explain why these dimensions coincide under two-sided Gaussian heat kernel estimates and discuss their relationships with other notions of dimension. We also introduce the energy image density property, its recent resolution, and its applications to martingale dimension. Finally, we show that spaces carrying diffusions with two-sided sub-Gaussian heat kernel estimates of walk dimension greater than two are not Lipschitz differentiability spaces.