Papers

We establish the full version of a conjecture of Guth and Zahl, giving a lower bound for the volume of a semialgebraic set that has a large intersection with a collection of $\delta$-separated $\delta$-tubes. Our proof uses o-minimal methods to simplify the proof of Katz and Rogers, who proved the conjecture up to a small factor. We also establish that the constants depend polynomially on the complexity of the semialgebraic set, and more generally in the #o-minimal setting.

Let $K \subset \mathbb{R}^N$ be a convex body containing the origin in its interior. In this work, we study the diameters of random sections of $K$ and derive upper and lower bounds for them in terms of geometric parameters of $K$. Our bounds hold with large probability, and they offer new insights into this widely studied subject. Our upper bound complements the so-called low $M^*$-estimate and in many cases it is much sharper. The two lower bounds that we give are each of a different nature: depending on the body in question each time, either could be better, and in many interesting cases it matches the upper bound too. Subsequently, we apply our results to determine random diameters of $p$-ellipsoids (images of $\ell_p$ balls under diagonal operators), improving upon previously known results and achieving sharp estimates in many cases. One notable application is to Information-Based Complexity Theory, where we manage to establish a simple (and essentially optimal) dichotomy in response to a very natural conjecture posed by Hinrichs, Prochno and Sonnleitner in 2023. Our solution settles precisely when it is useful to replace the optimal information used for the recovery of vectors from a $p$-ellipsoid with random (Gaussian) information, which can be more practical to obtain.

Let $C$ be a pointed closed convex cone in $\mathbb{R}^n$ with nonempty interior, and let $S^{n-1}$ denote the unit sphere in $\mathbb{R}^n$. The $L_p$ Minkowski problem for $C$-close sets is to determine, for a real number $p$ and a nonzero finite Borel measure $\mu$ defined on $\Omega_{C^\circ}=S^{n-1}\cap \mathrm{int} C^{\circ}$, whether there exists a $C$-close set $\mathds{A}$ such that $\mu$ is the $L_p$ surface area measure of $\mathds{A}$. In this paper, we will solve the problem for $p\in (0,1)$ and for $\mu$ being a nonzero finite Borel measure on $\Omega_{C^\circ}$. Moreover, we establish the continuity of solutions to the $L_p$ Minkowski problem for $p\in [0, 1]$ in several settings.

We prove a PL bordism version of Gromov's linearity conjecture over a broad class of asphericalization groups for manifolds endowed with faithful representations in all dimensions. Since every group embeds into an asphericalization group, this bordism version of the conjecture holds after allowing the target group to be enlarged. Our results apply in both the oriented and unoriented settings. In the PL category, no bounded local geometry is required. As an application, we obtain improved linear bounds for the Cheeger–Gromov $L^2$ $\rho$-invariants of PL $(4k-1)$-manifolds associated with their universal covers, replacing the previous superfactorial growth in dimension by factorial growth. The proofs are based on a quantitative asphericalization method that provides explicit control of the complexity of the resulting bordisms.

We show that the fractional illumination number of every convex body in $\mathbb{R}^d$ is at most $2^d$, with equality exactly for parallelotopes. We also prove that every such body can be covered by at most $2^d(d\log d+d\log\log d+O(d))$ smaller positive homothetic copies as $d\to\infty$, establishing the optimal exponential rate in Hadwiger's covering conjecture. The proofs use a covering measure obtained by minimizing an overlap energy and a greedy covering argument on a finite net.

We classify, up to quasi-isometry, the large-scale geometry of Galton–Watson trees for every finitely supported offspring distribution. Apart from the trivial finite diameter regimes, we condition on infinite diameter. We find that the remaining classes are the ray class, the full tree class (which includes the binary tree), one chain class $(\mathrm{C}_\Lambda)$ for every possible branching semigroup $\Lambda$, and the bushy class. Two independent trees almost surely admit a root-preserving quasi-isometry when their offspring distributions belong to the same class and are almost surely not quasi-isometric when they belong to different classes. Any two survival-conditioned supercritical realisations with finitely supported offspring laws nevertheless a.s.~admit quasi-isometric embeddings in both directions. We prove that the class of a realisation depends only on the support of the offspring distribution, not its specific distribution. For offspring distributions supported on $\{1,2\}$, we additionally obtain an explicit exponential tail bound for the probability of non-existence of a root-preserving $D$-quasi-isometry. The classification also implies that the corresponding random Cantor boundaries are almost surely quasisymmetrically equivalent. Conditioned on nonextinction, this applies to strongly separated fractal percolation, even when the underlying self-similar iterated function systems and retention parameters differ. We also classify two families of trees with continuous random branching times. Our proofs use new automorphism-matching theorems for random graph labellings of trees, based on contraction estimates for mismatch potentials. These matching results for Markov labellings on the binary tree may be of independent interest. The quasi-isometry classification, embeddability, and matching theorems are formally verified in Lean.

In this paper we find a sufficient condition on an Ahlfors–David regular metric measure space under which it admits a partition into parts of prescribed measures and optimal (up to a constant) diameters. The proof uses the construction of dyadic cubes. The process is algorithmic: the pieces are cut out one by one via a filling procedure on the tree of dyadic cubes. We introduce the notion of spaces which admit a connected dyadic cube decomposition and prove that they admit a partition of the kind described above. We then develop several techniques to obtain such spaces and show some natural examples of this type.

Convex spiral equicoverings were recently introduced by Espinosa-García, Martínez-Sandoval and Roldán-Pensado. They left open the question of whether every planar mass admits a convex $(3k,k+1)$-spiral equicovering. In this paper we give an affirmative answer to this question. To be precise, we prove the following: Given an integer $k \ge 2$, for every planar mass there is a fan consisting of $3k$ equal-mass sectors such that the union of every $k+1$ consecutive sectors is convex.

On the real line, Bárány, Kolossváry, and Troscheit [5] made exponential separation of analytic self-conformal systems checkable through a dual iterated function system on a space of analytic functions. The quantity their condition separates is the pre-Schwarzian derivative, and this observation lets us carry the construction to conformal iterated function systems in every dimension, where the pre-Schwarzian is $T_f=\nabla\log\|Df\|$. A pointwise separation of the pre-Schwarzian cocycle implies the strong exponential separation condition, and in dimensions one and two separation of the Schwarzian cocycle implies the same condition modulo Möbius maps. Each hypothesis is equivalent to a uniform gap condition, and these are $\mathcal{C}^2$-open. On the line they are also dense, so both separation conditions are $\mathcal{C}^2$-generic there; this sharpens the genericity theorem of Bárány, Kolossváry, and Troscheit, whose open and dense set carries only the plain condition. In the plane they are dense on Jordan domains with simply connected extension domains, and both separation conditions are $\mathcal{C}^2$-generic there too. In dimensions at least three every conformal map is Möbius, so no system satisfies the condition modulo Möbius maps, and the pre-Schwarzian hypothesis becomes a pole-separation condition, dense when the generators contract strongly enough and stay away from the similarities, but not dense in general. On Jordan domains, the planar genericity and a theorem of Feng and Rapaport [13] settle the dimension drop conjecture for $\mathcal{C}^2$-generic planar systems with injective generators.

Let $I_{\inf}(a):=\lim_{n\to\infty}I_{n}(a)=\inf_{n\ge1}I_{n}(a)$ be the asymptotic isoperimetric profile for product of a weighted Riemannian manifold satisfying $\mathrm{CD}(0,\infty)$ and, more broadly, for product of a nonsmooth subspace with density that can be approximated by a sequence of densities satisfying $\mathrm{CD}(0,\infty)$. We establish the following identity: \[ \lim_{a\downarrow0}\frac{I_{\inf}(a)}{a\sqrt{2\ln(1/a)}}=\sqrt{K_{\mathrm{LS}}}, \] where $K_{\mathrm{LS}}$ is the optimal log-Sobolev constant. The same argument also yields the large-deviation and moderate-deviation asymptotics for isoperimetry.

We prove that every three-dimensional parallelohedron $P$ satisfies \[ \mathcal H^2(\partial P)\ge 12\sqrt2\,r(P)^2, \] where $r(P)$ is the inradius of $P$, with equality if and only if $P$ is similar to the regular rhombic dodecahedron.

We generalize de Rham's decomposition theorem to proper metric spaces: every proper metric space has a unique decomposition as the product of a Euclidean space and a finite or countable pointed $\ell^2$-product of nontrivial irreducible non-Euclidean factors. More generally, every complete metric space splits canonically into a Hilbert factor, a pointed $\ell^2$-product of irreducible non-Euclidean factors, and a factor with no nontrivial irreducible factors. These results partially answer a question of Ulam.

We prove that the lattice covering density of the regular octahedron is $9/8$, settling a conjecture of Dougherty and Faber.

Among lattices normalized to have packing radius one, the known shallowest depths of the deep holes in $7$-, $8$-, $9$- and $10$-dimensional lattices are $\sqrt{7/3}$, $\sqrt2$, $\sqrt{5/2}$ and $\sqrt{8/3}$, given by $E_7^*$, $E_8$, the laminated lattices $\Lambda_9$ and $\Lambda_{10}$, respectively. This paper presents new lattices in these dimensions with shallower deep holes. It also answers two problems of J. Martinet about short representatives of the classes of $\Lambda/2\Lambda$ in $7,8,9,10$ and $12$ dimensions.

We prove that, for any $f:S^2\to S^1$ and any $\varepsilon>0$, there exist orthogonal vectors $x,y\in S^2$ such that the length of the shortest arc between $f(x)$ and $f(y)$ is at most $\pi/2 +\varepsilon$. This proves a conjecture of Ghebleh from 2007 that the circular chromatic number of the real orthogonality graph is equal to four.

We propose a computationally inexpensive algorithm to determine whether an opaque spherical body eclipses a luminous one, using only classical geometrical tools. We apply this method to the Sun-Earth-Moon system and predict solar eclipses, including their type, beginning, and duration, using the three-body problem. For predictions over a one-year period, the resulting errors are in the order of minutes.

We prove an exact isoperimetric inequality for cubes and tori under the $\ell_{\infty} metric. As a corollary, we recover the celebrated grid edge-isoperimetric inequality of Bollobás-Leader.

We show that if a sequence of expander graphs equi-coarsely embeds into a Banach space, then this Banach space fails Kasparov and Yu's Property (H). Consequently, no Banach space with Property (H) can be coarsely universal for all countable groups. We provide a Lean verification of our results.

We study the energy minimization problem for eight points on the unit sphere. For the logarithmic and Coulomb energies, we show that the unique global minimizer up to congruence is a square antiprism with height characterized by a unique stationarity equation. The proof is computer-assisted and fully verified in Lean. After this, we consider generalizations of the result to other important energies. For the Riesz $s$-energies, we provide a Lean-verified, non-computer-assisted proof that the square antiprism with height depending on $s$ is the unique global minimizer for all sufficiently large $s$, and a computer-assisted proof that this in fact holds for all $s\geq 0$. We also give examples of energies arising from completely monotonic potentials for which the square antiprism is not a global minimizer, answering in the negative a universality question of Cohn and Woo.

We derive explicit lower bounds for relative Fisher information by combining a variational principle with suitably orthogonalized Hermite-polynomial test functions. The resulting cumulant bounds are asymptotically sharp and yield lower bounds for Gaussian entropy deficits. We also establish quantitative versions of sharp min-entropy power inequalities in all dimensions. En route, we develop a stability result for Brzezinski's sharp bound for block sections of products of Euclidean balls, which may be of independent interest.

Moser's worm problem asks for a planar region of least area containing a congruent copy of every unit arc. We show that the infimum area $\alpha$ among convex universal covers satisfies $0.239\le\alpha\le0.24633\ldots$, reducing the gap between the previous refereed bounds by over $75\%$. For the lower bound, we choose four unit polygonal arcs and prove by finite subdivision that, however they are placed, their convex hull has area at least $0.239$. For the upper bound, we construct a quadrilateral of area $0.24633\ldots$ and prove cover universality by showing that its support inequalities force uncovered arcs to have length greater than one. The full proof is formalized in Lean 4 and verified by the Lean kernel. Code and certificates are available at https://github.com/ethan-keller/moser-worm-improved-bounds.

In this paper, we develop a polyhedral version of Milman's centro-affine spectral framework for full-dimensional origin-symmetric polytopes. A weighted graph Laplacian on the facets plays the role of the smooth centro-affine Laplacian. For $n\geq 3$, we prove that its first nonconstant even eigenvalue satisfies \[ \lambda_{1,e}(P)\geq n-1+\frac{1}{n-1} \] whenever $P$ is a local minimizer of $\lambda_{1,e}$ among origin-symmetric polytopes with the same facet normals, but not necessarily the same normal fan.

We determine, up to isometry, the Riemannian tangent and normal cones at arbitrary faces of boxes, acute orthocentric simplices, rectangular simplices, and asymmetric crosspolytopes in spaces of constant sectional curvature $\kappa\in\mathbb R$. Using these descriptions, we derive explicit one-dimensional integral formulas for their internal and external angles in arbitrary dimension. In particular, our results cover regular simplices, cubes, and crosspolytopes, the three families of regular polytopes existing in every dimension. We also establish a correspondence between regular simplices and cubes with isometric local cones at corresponding faces, provided their edge-length parameters are suitably matched. Combining our angle formulas with the Poincaré relation, we obtain volume formulas for simplices and crosspolytopes in even dimensions. For boxes, we derive volume formulas valid in every dimension. Among the new results are angle formulas for spherical and hyperbolic cubes and crosspolytopes, volume formulas for asymmetric crosspolytopes, and a one-dimensional integral representation for the internal angles of regular crosspolytopes that appears to be new even in the Euclidean case. The formulas involve the standard normal distribution function $\Phi(x)$ and its values at purely imaginary arguments $\Phi(\mathrm{i} x)$. For example, if $d\geq 2$ is even, then the volume of the ideal regular simplex in the $d$-dimensional hyperbolic space of curvature $\kappa=-1$ is given by the absolutely convergent integral $$ \frac{\pi^{d/2}} {\sqrt{2}\,\mathrm{i}^{d}\, \Gamma\left(\frac{d+1}{2}\right)} \int_{-\infty}^{\infty} \left[ \Phi\left(\frac{\mathrm{i} y}{\sqrt d}\right)^{d+1} + \Phi\left(-\frac{\mathrm{i} y}{\sqrt d}\right)^{d+1} \right] \mathrm{e}^{-y^2/2}\,\mathrm{d}y. $$

For finite measures with positive densities on convex Euclidean supports, we prove that $MCP(\kappa,N)$ passes with unchanged parameters to almost every isometric leaf of an arbitrary nonexpansive map. The proof rests on a sharp contraction inequality for geometric conditional densities, with exponent equal to the leaf codimension. The inherited dimension parameter is optimal. A total-variation limit on resolvent graphs extends the result to inverse fibres of maximal monotone relations, including convex gradient fibres. We also disprove Klartag's curvature-dimension inheritance conjecture by a firmly nonexpansive example in dimension three and a gradient example in dimension four. In codimension one, affinity of the geometric density yields curvature-dimension inheritance. The first example also gives failure on monotone fibres. Both constructions admit arbitrarily large curvature loss, including for a fixed Gaussian ambient measure on families of leaves of positive quotient measure.

Let $(\mathcal{M},d_{\mathrm{GH}})$ denote the Gromov-Hausdorff space of isometry classes of nonempty compact metric spaces. We prove that every nonempty compact metric space is isometrically embeddable into $(\mathcal{M},d_{\mathrm{GH}})$. More precisely, for every $D>0$ and every nonempty compact metric space $K$ with $\operatorname{diam} K\le D$, we realize the space of all $1$-Lipschitz functions on $K$ with values in $[0,D]$ as a family of metrics on a fixed Cantor space. Under this realization, the Gromov-Hausdorff distance agrees exactly with the uniform distance between functions, and each resulting metric space has diameter at most $76D$. We also construct finite approximations for which the Gromov-Hausdorff distance is given by an exact formula, together with a uniform approximation estimate.

We prove that every finite-dimensional Busemann \(G\)-space of dimension at least five has the disjoint disks property (DDP). For a sufficiently small metric sphere \(L=S(c,r)\), we show that every embedded arc contained in an exact distance level is a homotopical \(Z_2\)-set in \(L\). It follows that \(L\) has the disjoint arc-disk property and the disjoint homotopies property. Daverman's product theorem then gives DDP for \(L\times\mathbb R\), and a local avoidance argument at the center yields DDP for the ambient \(G\)-space. Since finite-dimensional Busemann \(G\)-spaces are generalized manifolds, in dimensions at least five the remaining obstruction to the Busemann conjecture is the resolution problem.

We address a problem on large signed sums of unit vectors that arose in work of Brugger, Fiedler, González Merino and Kirschbaum and was later formulated in its present form by Ambrus and Nietert. Given $d+1$ unit vectors $u_1,\ldots,u_{d+1}$ in $\mathbb R^d$, with $d\ge2$, the problem asks for the smallest possible value of \[ \max_{\varepsilon_i=\pm1} \left\|\sum_{i=1}^{d+1}\varepsilon_i u_i\right\|. \] We prove that this value is $\sqrt{d+2}$. We also determine all equality cases: up to independent sign changes and orthogonal transformations, they consist of the vertices of a centered regular simplex of positive even dimension together with an orthonormal basis of its orthogonal complement.

We prove that every finite Helly complex is isomorphic, at the level of its $1$-skeleton, to the Hellyfication (equivalently, the discrete injective hull) of its combinatorial boundary equipped with the metric induced from the complex. In particular, the boundary-rigidity phenomenon for Helly complexes proved by Blufstein-Chalopin-Chepoi admits a canonical injective-hull interpretation, independent of reconstruction procedures based on dismantling.

It is known that, for ordinary Hölder multifunctions with 0 < {\alpha} < 1, Hölder regularity alone does not guarantee the existence of a Hölder selection, and in general even a continuous selection may fail to exist. We ask how this situation changes when the parameter is measured through a Stieltjes clock. For 0 < {\alpha} < 1, we characterize the clocks for which every compact-valued g-Hölder multifunction admits a g-Hölder selection. The determining condition is uniform disconnectedness of the clock image Im(g). For left-continuous and nondecreasing clocks, this condition is equivalent to uniform jump dominance: every positive clock increment contains a jump carrying a fixed positive fraction of that increment. Under this condition, a selection can be chosen through any prescribed point of the graph, with explicit control of its Hölder regularity. Conversely, when the condition fails, there exists a compact-valued g-Hölder multifunction with no g-Hölder selection. The results show that the selection property is governed not simply by the presence of jumps, but by how their sizes are distributed across scales.

We show that the sharp Euclidean isoperimetric inequality holds for domains in complete simply connected Riemannian $n$-manifolds of nonpositive sectional curvature, $3\leq n\leq9$, which establishes the Cartan-Hadamard conjecture in these dimensions. The main step is a sharp inequality for constant-mean-curvature hypersurfaces, proved via integrals over pairs of boundary points, together with estimates for Jacobi fields along geodesic chords of the boundary. The weights in these integrals depend on the length of the chord and its angles with the boundary, and are chosen dimension by dimension, with a Green function pole. The inequality persists for boundaries of isoperimetric regions trapped in geodesic balls, whose mean curvature is constant only on the free part and which may have singularities. The isoperimetric-profile argument of Kleiner and Ghomi-Spruck completes the proof.