Papers

We prove that, for every integer $t\geq 1$, the unit two-sphere admits a spherical $t$-design consisting of exactly $54t^2$ points. More generally, such a design exists with exactly $6q^2$ points for every integer $q\geq 3t$. The proof builds on the topological degree method of Bondarenko, Radchenko and Viazovska, using an explicit equal-area partition of the sphere based on the map of Roşca and Plonka. By choosing the cell centers to minimize the average squared geodesic distance and deriving sharper sampling estimates, we obtain the stated quadratic bound on the size of spherical $t$-designs on $\Sph$.

Long-range expansion is a combinatorial graph property introduced to construct metric spaces with strong quantitative obstructions to low-distortion embeddings. Such extremal constructions are motivated by fundamental questions in metric geometry and nonlinear functional analysis. However, the only known constructions of long-range expander sequences are random regular graphs; the problem of constructing explicit sequences, especially with logarithmic girth, remains open. We resolve this question in strong form by proving that the Ramanujan condition implies long-range expansion. In particular, classical Ramanujan constructions (such as LPS graphs) give explicit long-range expander sequences with logarithmic girth. We also clarify that long-range expansion implies spectral expansion, and both implications are strict. While long-range expansion, the Ramanujan property, and spectral expansion are strictly separated, it is a priori not clear how robust these gaps are. We next compare them from a dynamics perspective: how do the different notions of expansion affect the mixing rate of random walks? Our second main result is that, from a dynamics perspective, LRE and the Ramanujan property are equivalent. We show that the random walk on any LRE sequence has the same cutoff location and Gaussian limit profile as on a Ramanujan graph. This incidentally offers intermediate progress between cutoff for Ramanujan graphs—proven by Lubetzky and Peres—and the long-standing conjecture of cutoff for transitive spectral expanders.

Talagrand asked for a construction of a large convex subset of a bounded Minkowski sum of a large Gaussian set. We first prove a stronger nonsymmetric existential statement: if $A\subset\mathbb{R}^n$ is measurable and $\gamma_n(A)>2/3$, then $A+A+A$ contains a compact convex set of Gaussian measure at least $1/2$. Let now $A$ be closed with $\gamma_n(A)\ge7/8$, let $\Phi$ be the standard Gaussian distribution function, and put $a_0=\Phi^{-1}(3/4)$. For every $\Lambda>1$ and every $0<p<2\Phi(\Lambda a_0)-1$ we construct a centrally symmetric finite polytope $C\subset\Lambda(A+A+A)$ with $\gamma_n(C)\ge p$. Thus measure $3/4$ is obtained for every $\Lambda>1.705510\ldots$. We give matching upper and lower bounds for the optimal high-measure dilation and show that the dilation profile used by the construction is sharp among all symmetric half-measure cores. Finally, if $\gamma_n(A)\ge5/6+\eta$, we construct, for every $0<\varepsilon\le1/2$, a centered ellipsoid $E_\varepsilon$ with $\gamma_n(E_\varepsilon)\ge1-\varepsilon$ and \begin{equation} \frac{c}{\Phi^{-1}(1-\varepsilon/2)}\sqrt{\frac{\log n}{n}}\,E_\varepsilon\subset A+A+A. \end{equation} Both the dimension dependence and the dependence on $\varepsilon$ are optimal up to constants. The construction also gives a six-summand theorem for balanced sets and a second finite construction based on subgaussian tests.

An equilateral set is a set of points in a metric space whose pairwise distances are all equal. Kusner conjectured that the maximum cardinality of an equilateral set in $\mathbb{R}^n$ with the $\ell_p$ metric is $n+1$ for every $1<p<\infty$. We disprove the conjecture for all $p>4$. In particular, for every such $p$, we define $m=m(p)$ and construct an equilateral set of $8m$ points in $\mathbb{R}^{8m-2}$. Previously, Swanepoel disproved the conjecture for $1<p<2$, while Ge, Xu, and Zhou showed that it holds for $2\le p\le 4$. Combining these prior results with the paper's main theorem resolves Kusner's conjecture for all $1<p<\infty$.

The infinitesimal Stoker conjecture states that an infinitesimal deformation of a convex polytope that preserves all dihedral angles also preserves all normal directions up to isometry. We give a short polyhedral proof, self-contained up to assuming spectral and geometric properties of the Izmestiev matrix.

Giuliano Basso asked whether every $n$-dimensional integral current space admits an integral current structure with the same characteristic set and multiplicity one almost everywhere. We give a negative answer already in dimension one. The obstruction is a parity phenomenon at odd-valence branch points. More precisely, if a one-dimensional integral current has unit multiplicity on the arms incident to an isolated vertex $v$, then the coefficient of its boundary at $v$ has the same parity as $\deg(v)$. Hence infinitely many odd-valence vertices force infinite boundary mass for every full-support unit-multiplicity current. We construct a compact geodesic, doubling, $1$-Ahlfors regular graph-like continuum $G$ of finite length and a boundaryless current $T\in\I_1(G)$ with $\set(T)=G$, while no $S\in\I_1(G)$ with $\set(S)=G$ can have multiplicity one $\Hh^1$-almost everywhere. We classify all full-support integral cycles on $G$. In particular, the least possible essential maximal multiplicity is exactly two, and the minimum mass of such a cycle is $3/2$. For finite metric graphs we identify a complementary sharp defect: the minimum boundary mass among unit-multiplicity currents equals the number of odd-degree vertices. Finally, by taking products with flat tori and using the precise Ambrosio–Kirchheim slice representation, we obtain boundaryless compact geodesic Ahlfors-regular counterexamples in every dimension.

We prove the Cartan-Hadamard conjecture (the sharp Euclidean isoperimetric inequality) in every dimension, assuming that the sectional curvature is pinched relative to a reference radius of curvature. The reference radius of curvature may vary, and the condition includes examples where the radius of curvature blows up, and the manifold has both unbounded (negative) and asymptotically vanishing sectional curvature. The key step in the proof is to control the entire convex hull of a quotient minimiser at its mean-curvature scale and compare the metric distortion with the surplus in the normalised hyperbolic isoperimetric profile at that same scale.

We study finite polyhedral models inside the projectivized space of equivalent norms $\mathcal{N}'(X)$ on a finite-dimensional Banach space $X$, endowed with the logarithmic distortion metric. Given a finite symmetric direction set $E$, we introduce the class of $\alpha E$-norms, defined as Minkowski functionals of symmetric polytopes whose vertices lie on the rays prescribed by $E$. We identify the admissible weights $\alpha$ for which this parametrization is non-redundant and prove that they give a one-to-one parametrization of the corresponding finite direction model $\mathcal{N}(E)$. We then refine this description by introducing complete, symmetric, simplicial $E$-fans, which encode the conical regions on which the associated polyhedral norms are linear. For a fixed fan $\mathcal{F}$, we define the corresponding geometric and coordinate fan models $\mathcal{N}(\mathcal{F})$ and $\mathcal{R}(\mathcal{F})$, and show that they are naturally isomorphic as cones. After quotienting by positive scalar multiplication, the coordinate models become isometric to the corresponding norm models: the logarithmic distortion metric on $\mathcal{N}'(E)$ and $\mathcal{N}'(\mathcal{F})$ is represented exactly by the Hilbert projective metric on the admissible weight spaces. Finally, we prove completeness results for these finite models and show how they provide finite-dimensional polyhedral approximations to the metric geometry of $\mathcal{N}'(X)$.

We study the spherical statistics of all the points in a thin shell of a high-dimensional random lattice. The exact relations between lattice points make it unclear when predictions based only on spherical geometry should hold. We answer this question for several statistics, including the number of shell points, the balance of their directions, and the occurrence, repetition, and distribution of differences between them. We identify sharp thresholds as the shell radius grows and show that these statistics undergo several distinct phase transitions. A shell can already agree with one geometric prediction while still differing strongly from another. Our results hold for a single sampled lattice, with probability tending to one as the dimension grows. They include estimates that hold across a complete shell, bounds close to the transition thresholds, and extensions to randomly shifted lattices. A Lean formalization verifies the main results, assuming the classical formulas and probability model stated in the code.

Let ${R^\ast}(n)$ be the least radius of a central circle to which nonoverlapping circles of radii $1,\ldots,n$ are externally tangent. We prove that ${R^\ast}(n)={C_\ast} n^2+o(n^2)$ and characterize ${C_\ast}$ by finite linear programs with an explicit error tending to zero. The reduction preserves arbitrary orders and all pairwise constraints: the limiting problem places marked points on a line at pairwise separation at least the geometric mean of their marks. Concatenation with a bounded boundary cost proves existence, and balanced finite-word programs supply matching effective upper and lower bounds. Their certified gap is $(1/k+1/r)/\pi$, before directed arithmetic error, for $k$ mark types and words of length $r$. A quantitative reflected-block recovery theorem supplies genuine permutations and full ring geometry, with a countable extension and a strict four-block improvement. The explicit interval is $C_{\mathrm{term}}+\eta_{\mathrm{width}}\le{C_\ast}\le U_4$; neither endpoint is asserted sharp. In particular, the coefficient $1/8$ proposed in the preceding finite study is false. An elementary expression for ${C_\ast}$, efficient high-precision evaluation and global floating-circle structure remain open.

We prove the existence of solutions to the Lorentzian Monge problem on synthetic Lorentzian spacetimes that satisfy the forward timelike measure contraction property $TMCP^+(K,N)$. The cost for this problem is the time separation function $\ell(x,y)$ (the Lorentz distance), which represents the maximum amount a particle can age when traveling from $x$ to $y$. The solution is based on the needle decomposition technique, and amounts to a reduction of the full problem to its one-dimensional counterparts. As an application of needle decomposition, we show that the Lorentzian timelike measure contraction property $TMCP^+(K,N)$ is equivalent to the $TMCP^+_{rLip}(K,N)$ condition, where the curvature dimension conditions are defined on gradient flow curves of reverse 1-Lipschitz functions.

For all prime powers $q\geq5$, we construct lattices $\mathcal{L}_q\subseteq\mathbb{Z}^q$ with kissing numbers \[ \tau(\mathcal{L}_q)\geq \left(\frac{1}{2\pi e^2}+o(1)\right)\sqrt{q}\,e^{2\sqrt{q}}. \] The same asymptotic bound holds on a set of integer dimensions of natural density $1$, and in every sufficiently large integer dimension $n$ with an additional factor $e^{-\tfrac{1}{2}n^{1/40}}$. The construction is an extension of a previous construction by Bennett-Peikert based on Reed-Solomon codes.

The Petty conjecture asserts that the quotient \(V_n(\Pi K)/V_n(K)^{n-1}\) is minimized among all convex bodies precisely by ellipsoids. In this paper, we prove the conjecture for all convex bodies of revolution in \(\mathbb R^n\), \(n\geq3\), including the equality cases.

We develop a geometric framework for distance-based formation control that separates the evolution of inter-agent distances from its realization by compatible node motions, reducing the stability problem to the edge space. We show that local exponential convergence of the edge dynamics implies local exponential convergence of the formation, and that stability is certified by spectral properties of a linear edge operator. We introduce a hierarchy of generic spectral properties — weak admissibility, admissibility, and strong admissibility — that provide necessary conditions for local exponential stability. Specializing to directed sensing, we obtain a necessary and sufficient spectral condition for local stability at an arbitrary target, together with a quadratic sufficient certificate. These conditions reveal that stability depends jointly on the graph orientation and target geometry, and show that persistence is neither necessary nor sufficient for local convergence. We show that every generically rigid graph admits an admissible orientation and, for acyclic orientations, we give an exact combinatorial characterization of admissibility. Finally, the quadratic certificate leads to a semidefinite program for synthesizing stabilizing edge gains.

We prove that minimal-volume hyperplane sections of the regular simplex through its centroid are parallel to a facet. The proof combines variational methods with Fourier-analytic techniques and zero-diminishing arguments to show that every critical normal vector has at most three distinct non-zero coordinates. Analysis of the two- and three-value cases then yields the sharp lower bound.

Let \(K,T\subset\mathbb R^n\) be convex bodies with the origin in their interiors. If \(K=T^\circ\), then the identity \( K+T=K^\circ+T^\circ \) holds trivially. V.~Milman asked whether these trivial solutions are the only ones, that is, whether \( K+T=K^\circ+T^\circ \) forces \(K=T^\circ\). We show that the answer is positive for polytopes but negative for general convex bodies.

Each of the 12 known conjecturally optimal kissing configurations in dimensions 5, 6, and 7 carries a contact-graph polytope: a polytope whose vertices are the configuration points and whose edges are exactly its contacts. Eight of these polytopes are non-convex and distinct from their convex hulls, which include longer edges. Each polytope is radial about the sphere's center: its boundary meets every ray from the center exactly once. It is the unique radial polytope with those vertices and edges. Seven of the nine non-lattice configurations carry polytopes constructed from the lattice contact polytope in the same dimension by splitting some facets and reassembling the pieces. This split-and-reassemble construction describes the configurations by facets rather than by layers. The fold angles of the reassembled facets explain the inner products $1/(10-n)$, giving $1/5$, $1/4$, and $1/3$ in dimensions 5, 6, and 7, respectively. The contact graphs of all 12 configurations are infinitesimally rigid. We also prove that $Q_5$ and $R_5$ are infinitesimally jammed, extending the 10 previously established cases to all 12 configurations. All mathematical computations use exact arithmetic.

We prove for the Euclidean space $\left(\mathbb{R}^n, \|.\|_2\right)$ the existence of subsets $A^n \subseteq \mathbb{R}^n$ with $d_H(\mathbb{R}^n, A^n)$ arbitrarily small, such that $d_{GH}(\mathbb{R}^n, A^n) = \sqrt{\frac{n+1}{2n}}d_{H}(\mathbb{R}^n, A^n)$, showing that the lower bound $d_{GH}(\mathbb{R}^n, A^n) \geq \sqrt{\frac{n+1}{2n}}d_H(\mathbb{R}^n, A^n)$ proved in arXiv:2607.18447 is tight for sets that are arbitrarily dense in the surrounding space. We also prove a similar statement for Riemannian manifolds and sufficiently dense subsets as a converse to arXiv:2309.16648.

We prove the Bárány–Kalai–Tverberg conjecture extending the affine version of Tverberg's theorem to simplicial balls and polytopes other than the simplex: For any affine map $\phi : P\to \mathbb R^d$ from a convex polytope $P$ of dimension $N=(d+1)(r-1)$ with $r\geq 2$, there exist $r$ pairwise disjoint faces $F_1,\ldots,F_r\subset\partial P$ such that $\phi(F_1)\cap\dots\cap\phi(F_r)\neq\varnothing$. A similar statement holds for a simplicial ball $P$ with $\phi$ assumed affine on all of its faces.

We characterize Gromov-compactifiability of Wasserstein spaces over Polish metric spaces for $1\leq p\leq\infty$. The space $\mathcal{P}_1(X)$ is Gromov-compactifiable if and only if $X$ is bounded and Gromov-compactifiable, whereas $\mathcal{P}_p(X)$ is Gromov-compactifiable for every $1<p<\infty$. At the other endpoint, $\mathcal{P}_\infty(X)$ is Gromov-compactifiable if and only if $X$ is. For $p=1$, the sufficiency argument combines an averaged triangle defect with tightness. For $1<p<\infty$, the main tool is the uniform convexity of $L^p$. The $p=\infty$ proof uses the finite-set criterion and a mass redistribution construction.

Fix one colour class in the checkerboard colouring of an $n\times n$ integer grid, and let $M_4(n,\varepsilon)$ be the largest subset having at most two points in every row, column, and diagonal of slopes $\pm1$. We prove the near-saturation bound $M_4(n,\varepsilon)\leq2n-4$ for $n\geq6$. The proof uses first and second moments of the four line families: a hypothetical set of size $2n-3$ produces row, column, and diagonal deficits whose exact moment identities contradict Cauchy–Schwarz. A finite argument handles $n=6$. The same identity extends to arbitrary deficit multisets. It gives $M_4(n,\varepsilon)\leq2n-d$ whenever $d\geq4$ is an integer and $n\geq3d-4$, and an entirely discrete asymptotic estimate \[ M_4(n,\varepsilon)\leq(\sqrt{21}-3)n+8. \] We also prove a general continuum-to-discrete theorem for the associated four-direction fractional packing problem. Applying it to the exact continuum dual certificate constructed in earlier work yields, for both colours, \[ L_{\mathrm{mono}}(n,\varepsilon)\leq\alpha n+O(1), \qquad \alpha\approx1.5768233968738, \] and hence the same upper bound for $M_4$ and for checkerboard no-three-in-line sets.

We prove Jensen's inequality on each coordinatewise parity class of a jump $M$-convex function on $\mathbb{Z}^n$, without assuming strong base orderability or a finite effective domain. If a finite convex combination of domain points has its barycenter in the same parity class, then the barycenter belongs to the domain and satisfies Jensen's inequality. We also prove that every domain point admits an affine support on its entire parity class. The set of support vectors is characterized by at most $3^n-1$ local inequalities, which yield optimality certificates, attained duality, and an exact formula for linear growth and stability under linear perturbations. The same conclusions hold for distributions with finite first moment supported on domain points of a single parity class, provided their mean lies in that class. We characterize equality by contact sets and give a quantitative bound on the Jensen gap. The proof combines induction on dimension with the known closure under infimal convolution: convolution with an even-box indicator gives finite-domain supports, a quadratic kernel proves barycenter feasibility, and a fixed local kernel yields global supports. A two-dimensional example shows that total parity alone does not suffice.

The problem of the Nile has been stated by R. Fisher in 1936. This problem concerns the fair division of an agricultural land and is the eponymous problem of a general class of problems on measures and similar regions. In 1938, W. Feller has shown that the general problem has no solution. However, we will show that under a natural hypothesis, the original problem of the Nile is solvable with connected plots.

We simplify Goodwin's construction of a non-compact closed convex hull in a CAT(0) space and show that three points suffice for the example.

We study a class of strongly local, regular Dirichlet forms on the standard unit interval. The aim of our work is to record some Hölder regularity properties that have not been noted in the prior literature. In particular, as our main result, we show that for every $\delta \in (0,1]$ there exists a metric measure space $(X,d,\mu)$ equipped with a strongly local, regular Dirichlet form $(\mathcal{E},\mathcal{F})$ on $L^2(\mu)$ with the property that $\delta$ is the supremum of $\alpha \in (0,1]$ for which the domain $\mathcal{F}$ of the Dirichlet form contains a non-constant $\alpha$-Hölder continuous function. To the best of our knowledge, such examples were previously known only for $\delta = 1$. In our construction, the value $\delta$ is characterized by the upper Hausdorff dimension of a certain Radon measure that is used to define $(\mathcal{E},\mathcal{F})$.

In the present work we study for a given Banach space the quotient of the family of equivalent norms modulus the collinearity relationship and prove that under a Banach-Mazur type metric this family is a complete metric space, and we also prove the completeness of the family of constructible norms for the Banach space of bounded Lipschitzian mappings between convex sets of a Banach space.

For every $n\ge4$ and $L>0$, we construct a smooth $4$-PIC metric on $S^n$ with Urysohn $1$-width at least $L$ and an embedded stable minimal disk of intrinsic inradius at least $L$. These examples disprove the proposed width and stable-disk radius bounds under a positive lower bound for isotropic curvature. On closed even-dimensional manifolds, we prove the sharp estimate $\lambda_1^{(2)}\ge(n-1)\sigma/2$ under $\sigma$-PIC and show that equality forces roundness if a closed eigenform attaining the bound has rank at least four at some point.

We study the spectral complexes arising from the de Rham complex on the Engel group $G$ viewed as a truncated multicomplex. We describe the two resulting complexes, analyse their compactly supported counterparts, and define the corresponding spectral currents by duality. We prove that the smooth spectral complexes embed naturally into the current complexes. Finally, we show that the Pansu pullback of a $W_{\mathrm{loc}}^{1,q}$ map $\varphi\colon G\to G$ with $q>7$ induces a morphism into the corresponding complexes of spectral currents.

We investigate the class of Liouville regular Jordan curves, which was introduced in our previous paper under the phrase "Jordan curves that admit continuous Legendrian lifts". We prove that a Jordan curve satisfying certain variation condition in a neighborhood of each point is in this class. We also prove that the boundary of a Jordan domain is Liouville regular if the images of the radial tails under a conformal mapping have uniformly vanishing length. Together with our previous result, these criteria imply the existence of an inscribed rectangle in every prescribed similarity class.

We show that the sharp Euclidean isoperimetric inequality holds for domains in complete simply connected Riemannian $5$-manifolds of nonpositive sectional curvature, which establishes the Cartan-Hadamard conjecture in that dimension. The main step is a sharp inequality for constant-mean-curvature hypersurfaces, proved via integrals over pairs of boundary points, in the spirit of Banchoff-Pohl, together with an estimate for Jacobi fields along geodesic chords. The inequality persists for boundaries of isoperimetric regions in geodesic balls, whose mean curvature is constant only on the free part. An isoperimetric-profile argument, after Kleiner, completes the proof. Our method also gives a new proof in dimension $3$.