Papers

We say that a finitely generated group equipped with a finite generating set is of type $\mathsf{R}$ if the Rips complex is contractible for sufficiently large scales. We show that the class of groups of type $\mathsf{R}$ is closed under taking graphs of groups with finite edge groups and, in particular, under taking free products. We prove that all two-dimensional right-angled Artin groups with the standard generating set are of type $\mathsf{R}$. Furthermore, we observe that products of finitely generated free abelian groups and finite groups are of type $\mathsf{R}$.

We develop a general theory for the dynamics of the geodesic flow of locally CAT(k) branched coverings and we show that it does not depend on the specific covering but only on the base space, the branching set and the degree of the covering. We find an explicit formula for the entropy: it equals the topological pressure of the associated natural dynamical system on the space of broken geodesics with a natural geometric potential. We find the exact asymptotic of the entropy as the number of sheets of the branched covering goes to infinity, as well as asymptotic properties of the measures of maximal entropy.

Let $K\subset\mathbb{R}^n$ be a convex body with support function $h_K$. For $\nu\in\mathbb{S}^{n-1}$, $p\in\mathbb{R}$, and $u\in\nu^\perp$, we introduce the slicing support function $h_\nu(u,p)$, defined as the support function of the slice $K\cap\{x\cdot\nu=p\}$ in the direction $u$. For each fixed $p$, this is precisely the support function of the corresponding translated fiber appearing in the construction of the convex fiber body of Mathis and Meroni \cite{MathisMeroni2023}. We derive an infimal representation of $h_\nu$ in terms of $h_K$, together with a corresponding minimax identity. Using the Fenchel–Moreau theorem, we prove that $h_K$, and hence $K$, can be recovered from the slicing support function without any regularity assumption on $\partial K$. We also obtain a differential recovery formula when $K$ is strictly convex and $\partial K$ is of class $C^1$. In dimension three, we establish a cylindrical Monge–Ampére-type determinant identity expressed in terms of the spherical curvature matrix of $\partial K$. When the relevant tangent directions are principal directions, this determinant reduces to a weighted ratio of the corresponding principal radii of curvature. We further characterize this principal-direction condition by showing that, for convex bodies with $C^2$-boundary and positive Gaussian curvature, the spherical coordinate directions are principal directions away from the poles if and only if, up to translation, the body is a body of revolution. Finally, we extend the construction to higher-codimensional iterated slicing support functions and derive a full-Hessian determinant identity via the Schur complement.

We study inequalities for the volume of orthogonal projections and their relation to Firey $L_p$-sum, together with their determinant-power analogues, motivated by the Dembo–Cover–Thomas conjecture. For $L_p$-zonoids $K,L\subset\mathbb{R}^n$ and $u\in S^{n-1}$, we consider the inequality \[ \left( \frac{|K\oplus_p L|} {|P_{u^\perp}(K\oplus_p L)|} \right)^p \geq \left( \frac{|K|}{|P_{u^\perp}K|} \right)^p + \left( \frac{|L|}{|P_{u^\perp}L|} \right)^p . \] For every $1<p<2$, we prove that this inequality fails in every dimension $n\geq2$. In contrast, the weak one-term inequality, obtained by omitting the second term on the right-hand side, holds in dimension two throughout the full range $1\leq p\leq2$. The proof of this planar result uses a sharp estimate for the normalized duality map. We also classify the corresponding determinant-power inequalities in the range $0<p<2$. The strong two-term inequality holds in dimension two and fails in every dimension $n\geq3$. The weak one-term inequality holds for $0<p\leq1$ in dimensions $n\leq3$ and fails for $n\geq4$; for $1<p<2$, it holds only in dimension two.

The classical Borell–Brascamp–Lieb inequality for multiple functions is an integral inequality relating finitely many input functions to a single output function. In this paper, we give an extension that allows a distinct output function for each input function. More precisely, we establish the following inequality. For a weight $ \lambda = ( \lambda_1 , \dots , \lambda_m ) $, we set $ z_\lambda ( x ) := \sum_{ i = 1 }^m \lambda_i x_i $ and denote by $ M_p^\lambda $ the weighted power mean of power $ p $. In addition, the power transform $ Q_d $ is defined as the continuous extension of $ p \mapsto p / ( 1 - d p ) $. We consider integrable functions $ f_1 , \dots , f_m , g_1 , \dots , g_m \colon \mathbb{ R }^d \to \mathbb{ R }_{ \geq 0 } $ satisfying $ 0 < \| f_i \|_1 < \infty $ for every $ i = 1 , 2 , \dots , m $. Then one has \[ \operatorname*{ess\,inf}_{ x = ( x_1 , \dots , x_m ) } M_p^\lambda \left( \frac{ g_1 ( z_\lambda ( x ) ) }{ f_1 ( x_1 ) }, \dots , \frac{ g_m ( z_\lambda ( x ) ) }{ f_m ( x_m ) } \right) \leq M_{ Q_d ( p ) }^\lambda \left( \frac{ \| g_1 \|_1 }{ \| f_1 \|_1 }, \dots , \frac{ \| g_m \|_1 }{ \| f_m \|_1 } \right) \] for any $ - \infty \leq p \leq 1 / d $, where the essential infimum is taken over the set of points satisfying $ f_i ( x_i ) > 0 $ for every $ i = 1 , 2 , \dots , m $. When all output functions $ g_1 , \dots , g_m $ are equal, this recovers the classical Borell–Brascamp–Lieb inequality for multiple functions. When $ p = 0 $ and the hypothesis is imposed pointwise, it coincides with a special case of the multi-output Prékopa–Leindler inequality of Cordero-Erausquin–Maurey.

For n points in the unit square, the Heilbronn triangle problem asks for the largest possible minimum triangle area. We develop a variational stress theory for this max-min problem. At every positive-area local optimum, normalized Karush-Kuhn-Tucker multipliers assemble into a skew matrix B satisfying Bz = 2ib, where b is the outward square reaction. This equilibrium has the isotropic affine moment sum_i p_i b_i^T = Delta I_2; the identity also holds for every tight subfamily carrying weights inherited from the same multiplier, with Delta scaled by its multiplier mass. It follows that every positive stress component meets all four sides and that there are at most two such components. To handle nonunique multipliers, we introduce the intersection of the stress kernels over the whole KKT face and a canonical hybrid operator incorporating all tight determinant and boundary derivatives. A strictly convex selector removes every decomposable invisible motion, leaving a rank-one-free residual with a sharp dimension bound; the literal maximal two-dimensional residual is excluded. Two-terminal substresses satisfy an exact interface-flux law, and a five-internal-vertex rank-four block has a Pfaffian cofactor carrier. Finally, an analytic reduction followed by exact symbolic enumeration excludes every one-external completion of a specified one-silent five-cycle residual, for all orientation words. These results isolate the remaining degeneracy but do not solve the problem for arbitrary n.

We exhibit 2-step Carnot groups with Dehn function $n^2 \log n$, sharpening Wenger's superquadratic lower bound for these groups. Some of these groups admit lattices, yielding finitely generated nilpotent groups with Dehn function $n^2 \log n$. This gives the first exact computation of a Dehn function which does not grow like $n^\alpha$ for any $\alpha \geq 1$ among nilpotent groups and, more generally, among finitely presented solvable groups with subexponential Dehn function.

For every integer $d\geq 1$, we construct a countably infinite set $X_d\subset\mathbb{R}^d$ in affine general position, with all pairwise distances rational. When $d$ is odd, $X_d$ may also be chosen so that no $d+2$ points lie on a common sphere. The construction is uniform in $d$: positive Chebyshev square decompositions produce harmonic curves on spheres whose points corresponding to rational parameter values have pairwise rational distances. A divided-difference factorization of the affine determinant shows that sufficiently short arcs are locally convex, and stereographic projection produces the odd-dimensional examples. We also construct infinite rational distance sets in $\mathbb{Q}^d$ in affine general position for every even $d$, and in general position for every $d\equiv 1\pmod 4$. For every $d\geq 1$ and $n\geq d+1$, taking and rescaling suitable finite subsets gives $n$-point integral point sets in affine general position. A suitable ordered choice yields integral-distance realizations of all cyclic polytopes. In dimension three, we give an explicit rational parametrization and obtain infinitely many pairwise non-similar primitive $n_3$-clusters for every $n\geq 4$.

Triangle-based measures, commonly referred to as badness functions, are widely employed to quantify the extent to which a distance matrix deviates from an ideal geometric configuration. Different formulations of these functions may capture distinct facets of local non-uniformity, and their behavior is often influenced by the underlying distance metric chosen for evaluation. In practical settings, although a canonical badness function may be conceptually preferred, factors such as computational cost, algorithmic constraints, or data-specific characteristics frequently necessitate the adoption of modified versions-for instance, approximate forms or alternatives defined under different distance metrics. This gives rise to a central question: to what degree do these variants retain the structural consistency properties of their original counterparts? To address this issue, we develop a systematic correlation-based framework for evaluating structural consistency. As an illustrative instantiation of this framework, we compute badness sequences from a set of representative distance matrices alongside randomly generated triangle configurations, which are designed to cover variants that may arise under diverse practical scenarios. We then assess pairwise similarities among these sequences using four correlation coefficients. The experimental outcomes indicate that certain badness variants exhibit a notably high degree of structural consistency, whereas others reveal complementary behavioral patterns; moreover, the choice of distance metric exerts a considerable influence on the observed trends. These findings offer practical insights for the informed selection of distance metrics and triangle badness function variants in tasks including geometric reconstruction, triangulation, and structural analysis of pairwise distance data.

The variance of nonparametric estimators is typically insensitive to the regularity of the object being estimated. We establish such a property for the spectra of graph Laplacian matrices at a fixed bandwidth $h>0$. Specifically, given $n$ i.i.d. samples from a probability measure $\mu$ on a Polish metric space, we compare the eigenvalues of the empirical weighted Laplacian operator $\Delta_{\mu_n}^h$ to those of the population counterpart $\Delta_\mu^h$ under a spectral gap condition, bounding the relative error by $1/\sqrt{nv_\mu(h)}$ for eigenvalues of order smaller than $h^{-2}$, where $v_\mu(h)$ is the smallest mass of a ball of radius $h$. This bound requires very weak regularity conditions on $\mu$: it is satisfied if $\mu$ belongs to the class of coarse PI measures that we introduce. This class contains measures on metric graphs, spaces with sufficiently regular boundaries, corners, or branch points, together with discretizations or thickenings of these at scale $O(h)$. Even for measures having densities of regularity $s>2$ on manifolds (the only known case so far), our bound improves on the state-of-the-art by shaving off logarithmic factors.

We consider families of pseudoconvex domains $X[s]\subset \mathbb C^n$ parametrized by complex manifolds $S$, and the maximal hermitian ellipsoids inscribed in each $X[s]$. The paper investigates how these maximal ellipsoids and their volume vary with $s$.

We give a short proof that for a given planar set of terminals $P$ there is at most one locally minimal tree with prescribed directions at $P$ (i.e. two locally minimal trees cannot coincide in $B_\varepsilon(P)$). The new ingredient is a combinatorial result which says that a certain embedding of a bipartite cubic graph has the same numbers of balanced and unbalanced vertices.

For $2 \leq a \leq d+1$, what is the largest integer $H_{a,d} (n)$ such that every set of $n$ points in $\mathbb{R}^d$ with no $a$ points on a common $(a-2)$-flat contains a subset of $H_{a,d} (n)$ points whose determined $(a-1)$-dimensional simplices have pairwise distinct $(a-1)$-dimensional volumes? We construct $n$-point sets that improve the best known upper bounds for $H_{a,d}(n)$ in several cases of $a$ and $d$. We also study a dual version of the problem. Let $D_d(n)$ the maximum number such that for any arrangement of $n$ hyperplanes in general position in $\mathbb{R}^d$, we can always find a subset of $D_d(n)$ hyperplanes for which all the $d$-dimensional simplices that they define have distinct $d$-dimensional volumes. We improve the current known upper bound for $D_d(n)$ and give the first nontrivial lower bound for $D_2(n)$ and $D_3 (n)$.

The Seven Circles Theorem states that if six circles form a closed chain and are tangent to a common circle, then the three lines joining opposite points of tangency on the common circle are concurrent. We extend this result to an arbitrary closed six-circle chain. Given such a chain, we consider two circles, each tangent to one of the two alternating triples of circles in the chain. For each circle in the chain, we then construct a circle tangent to it and to both of these circles. We prove that the three lines joining the centers of the newly constructed circles corresponding to opposite members of the chain are concurrent. In the classical configuration, the two circles associated with the alternating triples coincide with the common circle, while the six newly constructed circles degenerate to the six points of tangency, viewed as circles of radius zero. Thus, the classical Seven Circles Theorem is recovered as a degenerate case of our generalization.

For $n\geq 2$ and $1<p<n$, Ludwig-Xiao-Zhang \cite{LXYZ} asked whether the $L_p$-integral affine surface area of every convex body containing the origin in its interior is bounded above by the ball-normalized multiple of \(L_p\)-variational capacity, and whether the analogous analytic inequality holds for $L_p$-convexified level sets. We answer both questions negatively.

The dual $L_p$ John ellipsoids form a family that includes the origin-centered Löwner and Legendre ellipsoids. In this paper, we extend this family to general measures within the dual weighted $L_p$ Brunn–Minkowski theory. For measures with locally integrable densities that are continuous in a neighborhood of the boundary of the given star body, we establish a variational formula for $L_p$-harmonic radial combinations. The associated boundary integral leads to an optimization problem that remains meaningful under weaker assumptions on the boundary trace of the density. We prove the existence, uniqueness, and characterization of its solution. The resulting ellipsoids include the dual $L_p$ John ellipsoids for Gaussian measures. We also establish a monotonicity inequality, a measure inequality for homogeneous densities with its equality conditions on the weighted support, and an $L_p$ Löwner inclusion.

We prove stability estimates for the volume of central hyperplane projections of unit balls in $\ell_p^n(\mathbb{R})$, establishing the dual counterpart of the result of Chasapis, Nayar and Tkocz on stability of sections.

We establish variance lower bounds for the intrinsic volumes and the face numbers of random polytopes, generated by independent samples from rotationally invariant log-concave probability measures on $\mathbb{R}^d$ with full support. Our results extend the corresponding Gaussian lower bounds of Bárány and Vu and of Bárány and Thäle to this broader class of distributions. Our proof builds on the geometric construction introduced by Bárány and Vu, for which we develop an alternative treatment based on barycentric coordinates. We combine it with local variance estimates for both intrinsic volumes and the number of faces.

We describe GPT-5.6 Sol Pro's short proof that $2\cdot d_{GH}(S^1,S^{2k+1}) \le \tfrac{2\pi k}{2k+1}$. This is the odd-dimensional case of a tight upper bound by Harrison and Jeffs on the Gromov–Hausdorff distance between the circle and spheres.

Following a hint of Semmes, we employ Stong's bijections between integer lattices to construct space-filling surfaces, which are higher-dimensional analogues of space-filling curves. For each $m\geq 2$ we build $\alpha$-Hölder continuous parameterizations $f:[0,1]^m\rightarrow[0,1]^{m+1}$ with sharp exponent $\alpha=m/(m+1)$. In particular, there exist $(2/3)$-Hölder continuous surjections from squares to cubes. This solves Arnold's problem 1988–5.

We prove that any $n$-dimensional normed space is at Banach–Mazur distance $O(n^{2/3})$ from $\ell_\infty^n$.

Motivated by recent proofs of Gromov's codimension-two volume-growth conjecture, we propose a synthetic theory for scalar lower bound on metric measure Dirichlet spaces.

We prove that any Busemann G-space such that every sufficiently small metric ball is convex is a topological manifold. The key ingredient in the proof is Ivanov's Helly theorem. The appendix contains a counterexample to a question of Berestovskii–Halverson–Repovš.

For origin-symmetric convex bodies in $\R^n$ invariant under $n-2$ orthogonal hyperplanes reflections, we establish the logarithmic Brunn–Minkowski inequality and classify all equality cases without regularity assumptions. We derive the logarithmic Minkowski inequality and rigidity for cone-volume measures, including an explicit classification for bodies of revolution and uniqueness for simplicial polytopes in the prescribed symmetry class.

We describe a version of positive macroscopic scalar curvature motivated by the work of Alpert, Balitskiy, and Guth, and prove that this condition on a manifold implies a bound on its 1-width in terms of its first Betti number. A key tool in the proof is a decomposition of any closed manifold into a family of chains with convenient combinatorial structure, which was inspired by Nabutovsky, Rotman, and Sabourau's work on sweepouts. Finally, using techniques developed by Chodosh, Li, and Liokumovich, we show that for a sufficiently connected manifold, if the universal cover satisfies this curvature condition, then the manifold has a finite cover homotopy equivalent to $S^n$ or connected sums of $S^{n-1} \times S^1$.

We solve John Milnor's problem: among all convex regions of a given area on the sphere, the round disk requires the greatest distortion in cartographic projections onto the plane.

A valuation is a finitely additive function on the family of compact convex sets in $\mathbb{R}^d$. We study non-negative infinitely divisible random valuations, with particular emphasis on monotone, $\sigma$-continuous models with independent increments along nested families. After separating the deterministic part, we show that the Lévy measure of such a valuation is generated by pairs $(F,r)$, where $F$ is a non-empty closed convex set and $r>0$, with each pair contributing $r\mathbf{1}_{F\cap K=\emptyset}$. This yields a Poisson representation and an equivalent formulation through a pure-jump completely random measure on the space of closed convex sets. For stationary valuations, we derive a cylinder-Grassmannian representation of the Lévy measure. In the stationary isotropic case, we obtain a McMullen-type decomposition, at the level of one-dimensional distributions, into independent components stable under dilation of the argument.

Meissner polyhedra are constant-width bodies obtained from extremal finite sets of unit diameter. Such a generating set may contain dangling points, namely points having exactly two diametric neighbors. This article studies whether these points can play an essential role in surface-area minimization. Given an extremal set we show that the smallest surface area among the Meissner polyhedra based on it cannot increase by deleting a dangling point. Adding a dangling point cannot decrease the smallest achievable surface area. This reduces the search for area-minimizing Meissner polyhedra to generating sets without dangling points.

Let X be a metrizable space, and let Met(X) denote the space of metrics compatible with the topology of X, regarded as a subspace of the space of continuous pseudometrics with the supremum-metric topology. We first prove that if Z is a discrete space of cardinality aleph-one, then Met(Z) is not Borel. As a consequence, if Met(X) is Borel, then X is separable. Combined with a theorem of Koshino, our result yields that the space of bounded compatible metrics on a metrizable space X is completely metrizable if and only if X is sigma-compact. We also establish non-Archimedean analogues for spaces of ultrametrics.

The moving sofa problem asks for the largest planar set that can pass through a unit-width right-angled hallway. Gerver's 1992 construction remains the largest explicit example, but its defining switching parameters, continuum of supporting hallways, and contact topology involve transcendental equations and global inequalities. We give a reproducible exact-rational certification of the construction. A strict four-dimensional Krawczyk inclusion encloses the reduced switching system, while an independent twenty-two-dimensional Krawczyk calculation verifies Romik's full matching and contact equations in an explicit rational box. Exact interval propagation through the five analytic phases proves the support identities, global half-plane constraints, inner-wall separation, niche topology, oriented contacts, injectivity, and area bounds. From the certified support functions we identify every supporting hallway as L_t=x(t)+R_tL. For G=K_0\setminus\mathcal N(K_0) we prove directly that G is nonempty, closed, and connected and that \Phi_t(q)=R_{-t}(q-x(t)) is a continuous path in SE(2) carrying G through the hallway, with the correct initial and final arms. We further identify G with Romik's reconstruction of Gerver's classical eighteen-piece sofa. The deterministic companion verifier uses only Python's standard library and exact rational arithmetic. As a consequence, the Gerver-specific geometric hypotheses in Baek's recent variational framework are rigorously certified; we also clarify the one-sided tangent-speed convention at two contact-curve junctions.