Papers

We study the local metric completion of positive singular conformal metrics near a codimension-two corner when the conformal factor is generated by a compact positive measure on exponent space. The Newton support function determines noncritical weighted-ray accessibility, but at the critical power order support geometry alone is insufficient. We identify the missing invariant as a Laplace transform measuring how exponent mass approaches the critical total-degree face. If the maximal total degree exceeds 2, the local completion fibre over the corner is empty. When all total degrees are at most 2, the fibre is empty or a singleton according as the square root of the critical Laplace profile is nonintegrable or integrable at infinity. We give support-identical measures with opposite critical completion behaviour, sharpness examples, monotone truncation phenomena, intrinsic Lipschitz completion coordinates, and a bi-Lipschitz transfer principle for Newton-comparable densities.

Karasev proved, in dimensions two and three, a sharp surface-area counterpart of a polyhedral extension of Vaaler's theorem [8, Theorem 1.2]. We remove the dimension restriction. More precisely, if an $n$-dimensional convex polytope $P$ contains the origin in its interior and the affine hull of every nonempty proper face of codimension $k\in\{1,\ldots,n\}$ is at distance at least $\sqrt{k}$ from the origin, then $$ \mathcal{H}^{n-1}(\partial P)\geqslant n2^n. $$ Consequently, the boundary of every $n$-dimensional linear section of the cube $[-1,1]^N$ has $(n-1)$-dimensional measure at least $n2^n$. This confirms, in all dimensions, a conjecture of Grigory M. Ivanov recorded by Karasev [8, Section 1]. The proof combines the Rogers–Karasev flag decomposition with a dimension-free Gaussian comparison for orthoschemes. Its main step is an ordered-square substitution which converts all face-distance assumptions into a pointwise domination of a single positive integral. We also disprove the naive extension to all skeletal measures, formulate a ridge-skeleton conjecture, and establish two partial results toward it.

We study volume comparison for closed Riemannian manifolds satisfying a positive Ricci curvature lower bound together with an improved scalar curvature lower bound. We prove that if a closed Riemannian manifold $(N^n, g)$ satisfies $\operatorname{Ric}_g\ge (n-1)g$ and the scalar curvature $R_g\ge n(n-1)(1+\varepsilon)$, then its volume satisfies $$\lvert N\rvert_g \le \frac{1}{\sqrt{1+n\varepsilon}}\lvert\mathbb S^n\rvert. $$ In fact, assuming only $\mathrm{Ric}_g\ge(n-1)g$, we can prove that $$\frac{|N|_g}{\left|\mathbb{S}^n\right|} \le \frac{1}{|N|_g} \int_N\left(\frac{R_g}{n-1}-(n-1)\right)^{-\frac{1}{2}} d \mathrm{vol}_g. $$ The equality holds if and only if $N$ is isometric to the unit sphere. The proof combines a coefficient-adapted Jacobian comparison with a new integral shuffling comparison for scalar Jacobi solutions, inspired by Brown and Freedman \cite{BrownFreedman2022}. This yields an estimate that retains the full Ricci spectrum. The resulting volume bound agrees to first order in $\varepsilon$ with the factor appearing in Bray's conjecture. A further consequence of the argument is an averaged volume comparison for metric balls involving the scalar curvature. We also obtain a volume comparison theorem under a weighted integral lower bound on the Branson $Q$-curvature.

We show that any measurable, translation and $\mathrm{SO}(n)$-invariant valuation on polytopes in $\mathbb{R}^n$ is a linear combination of the intrinsic volumes, which extends Hadwiger's classical characterization of rigid motion invariant continuous valuations on convex bodies. This result is based on a novel regularity result for translation invariant, measurable, $1$-homogeneous, and simple valuations. As a consequence, we obtain a similar characterization of the Steiner point map on polytopes.

We show that for a discrete isometry subgroup acting on a proper CAT(-1) space X, if the critical exponent is less than $1$, then the critical exponent equals the Hausdorff dimension of the entire limit set. Consequently, the limit set must be a Cantor set. As an application, we prove that any finitely generated, torsion-free discrete subgroup in Isom(X) with critical exponent less than one must be geometrically finite and free. This answers a question of Kapovich.

Let $g:\mathbb{D}\to\mathbb{D}$ be a hyperbolic holomorphic self-map of the unit disk with Denjoy–Wolff point $\tau\in\partial\mathbb{D}$ and angular derivative $\alpha\in(0,1)$. We say that $g$ has an extremal hyperbolic rate if its forward iterates satisfy the sharp metric asymptotic \begin{equation*} \rho_{\mathbb{D}}(g^{\circ n}(z),w) = n\log\frac{1}{\alpha} + O(1) \quad\text{as }n\to\infty, \end{equation*} for every $z,w\in\mathbb{D}$. Using the Herglotz–Nevanlinna representation, we prove that $g$ has an extremal hyperbolic rate if and only if the associated boundary measure $\sigma$ satisfies \begin{equation*} \int_{\partial\mathbb{D}\setminus\{\tau\}} \log\frac{1}{|\zeta-\tau|} \,d\sigma(\zeta) < \infty. \end{equation*} We further show that this condition is equivalent to a non-degenerate angular asymptotic of the Koenigs linearization: for any conformal map $\phi_\tau:\mathbb{D}\to\mathbb{H}$ with $\phi_\tau(\tau)=\infty$, \begin{equation*} 0< \left| \angle\lim_{z\to\tau} \frac{h(z)}{\phi_\tau(z)} \right| < \infty, \end{equation*} where $h$ is a Koenigs function. We then extend the extremal-rate theory beyond the unit disk. For finitely connected hyperbolic planar domains, the disk characterizations transfer to ordinary boundary points of the associated deck transformation group. For holomorphic self-maps of the unit ball $\mathbb{B}^n$ and for $K$-quasiconformal self-maps, where no comparable Herglotz representation is available; we establish sufficient boundary regularity conditions that guarantee the extremal hyperbolic rate. The quasiconformal result is further extended to finitely connected planar domains at ordinary boundary points. These results show that, across the settings considered here, extremal hyperbolic growth is governed by the non-degeneracy of the boundary linearization at the Denjoy–Wolff point.

A theorem of Dougherty–Jackson–Kechris states that any equivalence relation generated by a single Borel function is hypersmooth. A well-known open problem is whether this can be generalized to equivalence relations generated by countable families of pairwise commuting Borel functions. We give an affirmative answer in the case where the functions are bounded-to-one. This generalizes the theorem of Gao–Jackson on Borel actions of countable abelian groups.

A unit-speed searcher starts at the origin of the Euclidean plane and must hit an unknown straight line whose direction and distance from the origin are both unknown. We prove that every deterministic search path has competitive ratio at least $C_{\log}\approx 12.5937096701246675$. The bound is unconditional: the path need not be cyclic, self-similar, spiral-like, or monotone in angle. For each projection direction, we compare the path with a zigzag obtained by sorting its alternating record turns. The resulting completion constraints are interpreted as jobs with scale-dependent deadlines and lower-bounded through a finite-window scheduling argument in logarithmic time. Averaging these directional bounds then uses the exact Euclidean velocity budget. In one dimension, the same method recovers the optimal cow-path constant $9$. Finally, Arb ball arithmetic provides a rigorous numerical enclosure of the constant.

The cluster graphs on $n$ vertices, the disjoint unions of complete graphs, have the integer partitions of $n$ as their isomorphism classes, and the quotient edit distance $q^*(\lambda,\mu)=\min_{\sigma\in S_n}|E(G_\lambda)\triangle\sigma E(G_\mu)|$ makes that set a metric space. Its geometry and its complexity both issue from one identity: $q^*$ is an affine function of the maximum of $\lVert X\rVert_F^2$ over the contingency tables with margins $\lambda$ and $\mu$. Our main result is an explicit optimal embedding. The weighted dyadic sums of the Ferrers staircase, taken at the critical exponent $\frac14$, give a map $F_n$ into $\ell_2^{\,<4n}$ that acts on a single partition and is computable in $O(n)$ time, and its distortion is $\Theta(n^{1/4})$. That order is optimal, since $c_2(\mathcal K_n)=\Theta(n^{1/4})$: the lower half follows from a $\Theta(\sqrt n)$-dimensional Hamming cube of partitions and Enflo's theorem, so the determination needs no other external input. The analytic core is a scale-free inverse inequality for every integer sequence with $v(1)=v(N+1)=0$ and $v(s)-v(s+1)\in s\mathbb Z$: its critical dyadic energy is at least $\lVert v\rVert_1^2/(63504\sqrt{\mathrm{TV}(v)})$. Combinatorially the same identity yields two explicit $\ell_1$ models, the vertex-mass metric on sorted degree sequences with $\frac12\delta_1\le q^*<\frac32\delta_1$ and the block-energy metric with $q^*\le B\le2q^*-1$, both constants optimal; hence $c_1(\mathcal K_n)\le2$, and an $O(n\log n)$-time algorithm returns an alignment of cost below $2q^*$ carrying the certificate $q^*\in[\lceil(B+1)/2\rceil,B]$. Computationally, deciding $q^*(\lambda,\mu)\le Q$ is strongly NP-complete and admits no FPTAS, while the farthest alignment is polynomial-time solvable. The best constant in the inverse inequality remains open; an exactly solvable chirp family caps it at $\frac23$.

For each $d\in\N_{\geq 2}$ we construct a Delone set $Y$ in $\R^{d}$ for which every Lipschitz bijection from $Y$ to $\Z^{d}$ has a very irregular inverse. For example, the inverse fails to be Lipschitz, even at just a single point. Further, we clarify the relationship between several notions of regularity for bijections between Delone sets, studied in the literature.

Let $K$ be an origin-symmetric full-dimensional convex polytope in $\mathbb R^n$, $n\ge 3$. We show that, for $-n+3<p<1$, the normed space $(\mathbb R^n,\|\cdot\|_K)$ embeds in $L_p$ if and only if it embeds in $L_1$. The case of $p\le 0$ is understood in the generalized sense. In particular, an origin-symmetric full-dimensional convex polytope $K\subset\mathbb R^n$, $n\ge 5$, is an intersection body if and only if $K$ is the polar of a zonotope.

Following previous work on metric bisectors with respect to the Korányi metric, we study here bisectors in the Heisenberg group $\mathfrak{H}$ equipped with the Carnot–Carathéodory metric $d_{cc}$.

Let $B_1^n$ be the standard cross-polytope in $\mathbb R^n$, let $g_1,\ldots,g_m$ be independent standard Gaussian vectors in $\mathbb R^n$, and set $G_m=\operatorname{conv}{\pm g_1,\ldots,\pm g_m}$. For $m=n^3$ it is proved that $$ \mathbb P\left\{d_{\mathrm{BM}}(G_m,B_1^n)\geqslant c n^{5/8}(\ln n)^{-5/8}\right\}\geqslant 1-\frac2n $$ for a suitable absolute constant $c>0$. This independently improves the polynomial exponent $4/7$ in Friedland's preceding work. Independent concurrent work of Friedland, which appeared after completion of the present manuscript, obtains the same polynomial exponent with the stronger logarithmic factor $(\ln n)^{-1/4}$ by a different argument. The proof uses Friedland's discretization and conditioning argument together with the $K/U$ decomposition. A selected family of $K$ vectors is suppressed and the remaining $K$ vectors are quotiented out. In the resulting quotient simultaneous bounds are proved for every top-dimensional exterior product formed from the suppressed $K$ vectors and the $U$ vectors. A Dvoretzky-Rogers selection after L"owner normalization converts these determinant estimates into a bound for the minimum volume ellipsoid of the whole projected polytope and Maurey's empirical method then gives the required Gaussian measure estimate.

We establish the Moran–Hutchinson formula for attractors of finite systems of surjective similitudes on semimetric spaces. More precisely, for a complete, normal semimetric space satisfying strong regularity and geometric doubling, we prove that the open set condition implies that the Hausdorff measure of the attractor at the similarity dimension is positive and finite. We also prove the converse, specifically, positivity of the Hausdorff measure at the similarity dimension implies the open set condition. This provides a partial answer to a question posed by Bessenyei and Pénzes in 2022.

The lamplighter group over the planar integer grid is proved to not be bi-Lipschitz equivalent to any metric space of negative type, so in particular it does not admit a bi-Lipschitz embedding into $L_1$. This shows the existence of finitely generated metabelian groups on which word metrics are never comparable up to constant factors to conditionally negative definite (CND) kernels, and that the property of admitting a word metric-comparable CND kernel is not preserved by wreath products.

Atiyah's Minkowski-space version of the configuration-of-points construction assigns to an admissible marked configuration of $n$ worldlines a collection of $n$ binary forms of degree $n-1$, whose roots are the ordered retarded celestial directions. He conjectured that these forms are always linearly independent. We disprove this conjecture for every $n\ge3$. For $n=3$, an explicit planar one-parameter family yields a real coefficient determinant with exactly one simple zero in a specified interval. At this parameter, all six ordered celestial roots are distinct and the coefficient matrix has rank exactly two. A null-translation construction then multiplies the first three forms by a common factor and produces counterexamples for every $n>3$. Consequently, within the class of complete pairwise disjoint timelike affine lines, universal independence holds at $n=2$ and fails for every $n\ge3$; for each of the counterexamples, the normalized Atiyah–Sutcliffe determinant is defined and vanishes.

A small polygon is a planar polygon of diameter at most one; let $A_n$ be the largest area at order $n$. Using the global characterization of the even-order maximizers established in a companion paper, we determine their asymptotic geometry. After scaling the angular deficits near the unique pendant diameter, the exact critical equations converge to an autonomous second-order recurrence. Its marked boundary condition selects a unique positive half-line orbit, equivalently the unique minimizer of an explicit strictly convex action. The orbit approaches the regular state with stable multiplier $(-3+\sqrt5)/2$, producing an alternating, exponentially damped boundary layer. A uniform finite-cycle shadowing theorem transfers this profile to the true maximizers. For every fixed depth, an excursion-clipping argument proves that the positive variational finite section is the unique global minimizer on the limiting geometric domain of the Bingane–Mossinghoff construction; this conclusion is expressly distinct from minimization on a looser algebraic box. The sections converge sharply, with two-step error ratio $|(-3+\sqrt5)/2|^4$. The limiting constant $q_*$ has an exact variational definition and a certified rational enclosure. For even $n\to\infty$, $A_n=\frac\pi4-\frac{5\pi^3}{48n^2}-\frac{q_*\pi^3}{n^3}+O(n^{-4}).$ We also identify the leading gap from the Foster–Szabo upper bound and prove that $A_n$ is represented, up to an exponentially small error, by a real-analytic function of $1/n$.

We give a characterization of the existence of Lipschitz surjections between dust-like self-similar sets of the same dimension under the assumption that the similarity ratios are commensurable. Our results apply more generally to graph-directed fractals.

We disprove the conjecture that every tree with t leaves embeds isometrically into $\ell_\infty^{\lceil \log_2 t\rceil}$. We construct a 32-leaf tree whose least isometric $\ell_\infty$-dimension is six rather than five, and prove that every tree with at most 31 leaves attains the conjectured bound; Brigham et al. had recorded equality through 21 leaves. Thus 32 is the first failure, and the example answers affirmatively a question of Fitzpatrick and Nowakowski from 2000. The same topology has dimension six under every assignment of positive edge lengths, and therefore also disproves the later sharp leaf-threshold conjecture for weighted metric trees.

Let $X$ be an ultrametrizable space. We study the space of bounded compatible ultrametrics on $X$, equipped with its natural non-Archimedean distance. For every positive integer level, we prove that additive absolute Borel complexity of this moduli space implies multiplicative absolute Borel complexity of $X$ at the same level, and conversely. We also prove that an ultrametrizable space is a countable union of locally compact subspaces if and only if it is a countable union of closed subsets in every completion induced by a bounded compatible ultrametric. As a consequence, this moduli space is completely metrizable exactly when $X$ is a countable union of compact subsets.

In this paper we provide new constructions of equiangular tight frames of size $2d$ in $\mathbb{C}^d$. We generalize the doubling construction of Fallon and Iverson to a tensor multiplication construction based on a suitable pair consisting of a complex Hadamard matrix and an equiangular tight frame. In particular, such a pair always exists whenever there is an amicable pair of real Hadamard matrices. Most notably, amicable Hadamard pairs of order $q+1$ exist for all prime powers $q\equiv 3\pmod 4$. We also find specific constructions based on a family of pairs of order 6 and on pairs whose equiangular tight frames are defined by Paley conference matrices with $q\equiv 1\pmod 4$. Finally, we provide a power construction of equiangular tight frames that generalizes the construction of Turyn for conference matrices.

In the paper, we develop a structure theory for metric measure spaces with synthetic lower Ricci curvature bounds, known as RCD spaces. Under a positive injectivity-radius assumption, we recover a smooth differential structure and prove a regularity result: such spaces are $W^{1,p}_{\mathrm{loc}}\cap C^{0,\alpha}_{\mathrm{loc}}$-Riemannian manifolds whose weight functions also belong to $W^{1,p}_{\mathrm{loc}}\cap C^{0,\alpha}_{\mathrm{loc}}$, in both distance and harmonic charts for all $p<\infty$ and $\alpha \in (0,1)$. We also establish a quantitative lower bound for the harmonic radius, together with compactness theorems and quantitative geometric bounds. As a byproduct, we develop a comprehensive elliptic regularity theory in this setting. These results apply, in particular, to smooth weighted Riemannian manifolds and to RCD spaces satisfying a synthetic curvature upper bound, or CBA condition. Even in these settings, the resulting regularity statements are new. In the latter case, both the Riemannian metric and the weight function are shown to be locally Lipschitz. As further applications, we establish fibration theorems and use metric smoothing to confirm, in a synthetic framework, a conjecture of V. Kapovitch concerning almost flat manifolds with mixed curvature bounds.

In \cite {ChenFengLiXiXu2026}, a reverse projection inequality for general convex bodies was established in three dimensions. In this paper, we completely characterize the equality cases of this inequality. We further show that the reverse projection inequality fails in every dimension $n\geq9$. The counterexamples are constructed as full-dimensional polytopes with at most $n+2$ facets.

We obtain density Ramsey theorems for configurations consisting of the vertices of a simplex $\Delta_o$ together with their barycenter. We prove that any subset $A\subseteq\mathbb{R}^n$ of positive upper density contains an isometric copy of all sufficiently large dilates of $\Delta_o$ together with its barycenter. As this configuration is non-spherical such results are not possible with respect to the quadratic Euclidean metric, we consider general metrics $\rho$ defined by a positive-definite, homogeneous forms of even degree at least four. We prove the analogous result in the discrete setting, for subsets $A$ of the integer lattice $\mathbb{Z}^n$, under some natural and necessary congruence restrictions on the scales $\lambda$ at which the set $A$ can contain an isometric copy of the simplex.

We prove that every unimodular locally compact group $G$ of noncompact Lie dimension $n$ satisfies the sharp Brunn–Minkowski inequality \[ \mu_G(XY)^{1/n}\ge\mu_G(X)^{1/n}+\mu_G(Y)^{1/n}, \] and establish a general form for arbitrary, possibly nonunimodular, locally compact groups. This fully confirms the nonabelian Brunn–Minkowski conjecture proposed by the present authors and Zhang. As an application, we obtain an isoperimetric inequality on symmetric spaces of noncompact type.

Motivated by the conjectured monotonicity of the magnitude functions of metric spaces of negative type, we investigate the monotonicity problem for finite semimetric spaces, namely, “metric spaces without the triangle inequality”. We completely resolve this problem by proving that the monotonicity holds for negative type semimetric spaces with at most three points, while constructing negative type semimetric spaces with at least four points whose magnitude functions are not monotonically increasing. We also prove monotonicity for all 4-point metric spaces and construct 12-point metric spaces of negative type whose magnitude functions are not monotonically increasing.

A small n-gon is a planar n-gon whose diameter is at most one. For odd n, Reinhardt proved that the regular polygon is optimal. For even n, the maximizer is nonregular, and only a few low orders were known exactly. We prove that for every even n >= 8 the maximum-area small n-gon is unique up to Euclidean isometry and reflection. Foster and Szabo's description of the diameter graph reduces a maximizer to an (n-1)-cycle of unit distances together with one pendant diameter. We determine the compatible boundary order, interpret the cycle as the centers of a Reuleaux (n-1)-gon, and eliminate the pendant vertex by a one-variable area calculation. The remaining first-order equations have conserved translation and rotation quantities. In the resulting radial variables they become the critical-point equations of an explicit function on a convex domain. We prove that this function is strictly concave by factoring the relevant principal minors of its local Hessian. Compactness gives existence, while strict concavity gives uniqueness. The proof is analytic. The symbolic scripts supplied with the paper check algebraic identities but are not used as part of the proof. Combined with the classical odd-order and low-order results, this determines the maximal area for every n >= 3.

We study lower tails of support functions of isotropic convex bodies. An endpoint negative moment estimate, combined with a spherical cap argument, gives polynomial deviation estimates for the inradius of random projections in every dimension; in the unconditional class this yields the optimal order of the median inradius and shows that the cube is extremal. The same argument gives estimates for convex hulls of independent rotations and, after an additional projection, a mixed rotation-projection theorem. We also prove weighted lower and upper estimates for Minkowski averages of rotated polars and their random projections, as well as an inradius dependent estimate for the geometric distance of global averages from the Euclidean ball. A family of isotropic product cylinders shows that the latter estimate is sharp, up to absolute constants, throughout the possible range of the inradius. For the cube, an explicit negative moment computation gives a quantitative projected $(m,k,n)$-profile and, in full dimension, recovers the classical order $n/\ln(en)$ for bounded geometric distance. Finally, we determine the sharp volume profile of intersections of independent rotations in the unconditional isotropic class and show that the cube is extremal.

In a high-dimensional Funk ball, the endpoint potential makes the forward and reverse distances behave differently, while symmetrization erases the distinction that survives in pyramid limits. For Euclidean beta-type radial measures, we determine the natural-scale limits in all three parameter phases. The divergent phase yields directed Gaussian pyramids indexed by the balance between dimension and radial concentration. A positive limiting parameter yields a Gaussian–chi Funk horocone pyramid, built from Gaussian bases, chi-distributed heights, and a directed endpoint-potential increment, while a vanishing parameter yields the universal directed star pyramid. In the last phase, the dimension-dependent logarithmic contributions cancel exactly, so the conclusion requires no further rate condition.

In this paper we study Borel graphs generated by finitely many commuting Borel functions. We give a geometric analysis of the free part of such graphs based on marker sets and marker regions. Assuming the existence of $r$-forward-independent hitting sets with bounded syndeticity, we obtain marker decompositions of the free part into rootless and rooted regions with controlled geometry. As applications, we derive finite Borel asymptotic dimension and hyperfiniteness, and obtain upper bounds for Borel edge chromatic numbers which improve previously known results. For the case in which each of the commuting Borel functions is bounded-to-one, we verify the existence of $r$-forward-independent hitting sets with syndeticity $Cr$ for some constant $C$. This gives another proof of a recent theorem of Shinko-Weilacher-Yu, and is used to show that if one of the commuting Borel functions is injective and another one is bounded-to-one and exactly even-to-one, then the graph has a Borel perfect matching.