Papers

For finite metric spaces, magnitude is an invariant known to be discontinuous everywhere with respect to the Gromov-Hausdorff distance. While much of the literature focuses on positive definite metric spaces, here we consider finite metric spaces admitting a nonnegative weighting. In this paper, we focus on the class of finite metric spaces that admit a nonnegative weighting. This class includes ultrametric spaces, for which specific bounds on magnitude are known. We generalize these bounds by proving that, for any two finite metric spaces with nonnegative weightings, the ratio of their magnitudes is exponentially bounded by their Gromov-Hausdorff distance. Consequently, we show that within this class, the logarithmic magnitude function is Lipschitz continuous, and the magnitude function itself is locally Lipschitz continuous, but not Lipschitz continuous.

Let $1\leq k\leq n$. We prove that $k$-dimensional intrinsic Lipschitz graphs in the Heisenberg group $\mathbb{H}^n$ satisfy a geometric lemma $\mathrm{GLem}(\beta_{2,\mathcal{V}_k},p)$ for horizontal $\beta$-numbers with an exponent $p=p(k)$. Previously, this result was known only in the case $k=1$; our proof recovers the sharp exponent $p=4$ in this setting. For $k>1$, we adapt an integral geometric approach originally developed by Orponen for Euclidean and parabolic Lipschitz functions. In addition, for $k=n$, we show how to deduce a geometric lemma with $p=4$ directly from an isotropic Dorronsoro theorem in $\mathbb{R}^{2n}$ using a Poincaré inequality. Building on the new geometric lemmas, we establish a necessary condition for $k$-regular sets in $\mathbb{H}^n$ to admit corona decompositions by intrinsic Lipschitz graphs. The condition is known to be sufficient by earlier work of the last two authors together with Pinamonti. It involves additional flatness coefficients besides $\beta_{2,\mathcal{V}_k}$. Along the way, we therefore extend the known stability results for geometric lemmas under the “big pieces” functor to a larger class of coefficients.

We investigate Sylvester's classical four-point problem for ball-convex bodies, where convex hulls are replaced by intersections of balls of a fixed radius $R$. We show that the Efron–Buchta identities persist in this framework, so that the distribution of the number of vertices of the $R$-ball-convex hull of $n$ uniform random points is determined by the moments of the volumes of the hulls of its subsamples, and vice versa. For three and four uniform random points in an arbitrary planar ball-convex body, we determine the Sylvester probabilities of the $R$-ball-convex hull. We also give an extension of Groemer's inequality for the expected area of the ball-convex hull of $n$ random points and show that it is minimised by the disc of the same area. In particular, this yields an isoperimetric inequality for the ball-convex analogue of the affine length of the boundary curve. For the unit disc, we can express the Sylvester probabilities using dilogarithmic functions of the radius $R$, and give explicit values at $R=1$. As $R\to\infty$, these recover the classical Sylvester probabilities. As another consequence, we determine the exact distribution function of the circumradius of three uniform random points in he disc on $[1,\infty)$.

We study the relationship between the Cramér transform and Tukey's half-space depth for log-concave probability measures. For the uniform probability measure $\mu_K$ on a convex body $K\subseteq\mathbb{R}^n$, we prove the sharp pointwise comparison $$\Lambda_K^*(x)\leq -\log q_K(x)\leq \Lambda_K^*(x)+\frac12\log n+C,\qquad x\in\operatorname{int}(K),$$ where $C$ is an absolute constant. The order $\log n$ is optimal, as shown by the Euclidean ball. The proof combines exponential tilting, one-dimensional log-concavity, and self-concordance of the Cramér transform. As consequences, we obtain sharp-order moment and tail estimates for $\Lambda_K^*$ and identify $\exp(\Lambda_K^*(x))$, up to polynomial factors in the dimension, with the number of independent samples needed for $x$ to be captured by their random convex hull. We also establish an $O(n^2)$ variance bound for the logarithmic half-space depth and use it to derive a general criterion for sharp thresholds of random convex hulls. In particular, this criterion applies to the uniform measures on $\ell_p$-balls for every $p>1$. These results establish a quantitative link between large-deviation cost, geometric depth, and sampling complexity in high-dimensional convex geometry.

In this paper, we extend a large part of the uniformization theory of Bonk-Heinonen-Koskela [Asterisque 2001] to length spaces that are not necessarily proper or geodesic. Among other things, we show that there is a one-to-one correspondence between the quasiisometry classes of complete roughly starlike Gromov hyperbolic spaces and the quasisimilarity classes of bounded uniform spaces, which provides an affirmative solution to an open question of Bonk-Heinonen-Koskela. Our approach relies crucially on the work of Väisälä [Expo. Math. 2005], who investigated in depth Gromov hyperbolic spaces that are not necessarily proper or geodesic. One key new ingredient is to use the so-called (quasihyperbolic) $(c,\mu)$-quasigeodesic as a suitable substitute for quasihyperbolic geodesic.

We study the simultaneous Busemann-Petty and Shephard volume comparison problem: whether comparison of the volumes of all central hyperplane sections and all orthogonal hyperplane projections determines the ordering of the volumes of two convex bodies. For every $n\geq5$, we construct origin-symmetric convex bodies of revolution $K,L\subset {\mathbb R}^n$ such that every central hyperplane section and every orthogonal hyperplane projection of $K$ has strictly smaller volume than the corresponding section or projection of $L$, while $|K|>|L|$. For $n\leq4$, the affirmative solution of the Busemann–Petty problem shows that the section inequalities alone imply $|K|\leq|L|$. Without origin symmetry, we construct such counterexamples in every dimension $n\geq2$, with one body a nontrivial translate of a Euclidean ball and the other a noncentrally symmetric body of constant brightness.

We prove a sharp higher-order Rademacher-chaos inequality on the Hamming cube using random-reveal martingales. As a consequence, we obtain the sharp metric $X_p$ inequality for $L^p$, resolving an open question posed by Naor.

We study online search for an unknown affine hyperplane in $\mathbb{R}^D$, for arbitrary fixed finite dimension. Building on a companion self-similar cell reduction and support-function formulation, we ask how the mechanism changes as the normal space grows from $\mathbb{S}^0$ to $\mathbb{S}^{D-1}$. In $D=1$, alternation and productivity yield an equal-ripple principle and the exact stationary constant $9$. In $D=2$, the analogous relative equilibrium is a logarithmic spiral whose bottleneck chord imposes tangency and selects the pitch. For exponential orbits $\Gamma(\sigma)=e^{\kappa\sigma}\omega(\sigma)$, we develop log-directional geometry, exponentially discounted memory, gauges, and recursive hyperspherical parametrizations. Without a shape ansatz, the bottleneck admits a certificate supported by at most $D$ historical suppliers, and at globally worst phases the current point lies on the active face. Within regular chambers we derive exact variation, tangency, pitch, age, and, in $D=3$, delay-system identities. Odd-dimensional obstructions, antipodal subclasses, and harmonic towers provide constraints and explicit candidate families but are not claimed globally optimal. Finally, the N-COMP theorem shows that $C_D^*$ is a computable real for every fixed finite $D$ and that algebraic polygonal $\varepsilon$-optimal cells can in principle be synthesized. Numerical screening through $D=10$ is kept separate from the proved results.

A shoreline-search path starts at the origin and must meet an unknown affine line, without knowing either its normal or its distance. We first establish a self-similar reduction theorem for homogeneous search problems whose historical information is a record profile updated by pointwise maximum. Two quasi-returns of the normalized state delimit a block that renews the required profile by itself; a short connector closes this block into a cell. Every finite-ratio path can therefore be approximated, with arbitrarily small loss, by repetitions of a single cell at all scales. The main chain is then made effective. A finite coding of the state space computably bounds the scale factor and normalized length of a nearly optimal cell. For planar Shoreline search, the support function of the convex hull gives an exact cell functional. A one-sided polygonalization then reduces the problem to a computable number of vertices, after which quantifier elimination decides whether a polygonal cell exists below a rational threshold. It follows that the optimal deterministic planar Shoreline value $C_2^*$ is a computable real: for every rational $\epsilon>0$, an algorithm terminates with a rational interval of width at most $\epsilon$ containing $C_2^*$. Additional results—sliding memory, Bellman transitions, deadlines, geometric filters, and relative equilibria—are presented separately as a toolbox for certified computation and for the study of spiral rigidity; they are not used in the computability proof.

A classical kaleidocycle is a closed cyclic chain of congruent, edge-hinged tetrahedra with successive hinge axes at right angles, capable of continuous inside-out motion. Kaleidocycles composed of six isosceles tetrahedra (disphenoids) form a natural generalization in which consecutive hinge axes need not be perpendicular. Their kinematics are governed by the closure equation for threefold-symmetric Bricard 6R linkages. We derive a new compact closure equation directly from threefold symmetry using the composition of rotations, and show its equivalence to the classical formulation. Furthermore, we establish two equivalent necessary and sufficient criteria for a kaleidocycle to admit continuous, unblocked motion in terms of the geometric parameters of the constituent disphenoid. These criteria yield sharp bounds on the admissible geometric parameters of realizable kaleidocycles. Finally, we study assemblies of eight such kaleidocycles joined face-to-face. We identify a configuration exhibiting full octahedral symmetry, which forms a polyhedral structure closely related to the disdyakis dodecahedron.

We prove that a positive-definite quadratic form has nonnegative conorms if and only if it is matroidal, resolving a conjecture of Dutour Sikirić and Kummer. We show that the positive conorm support determines a regular matroid, and the positive conorms are the coefficients of a unimodular integral decomposition of the form. As a result, we show that a positive-definite form lies in the tropical Schottky locus if and only if its conorms are nonnegative and its positive conorm support matroid is cographic. This answers a question of Chua, Kummer, and Sturmfels and provides algorithms for the tropical Schottky decision and recovery problems in every genus.

Let $\Gamma\subset\mathbb R^2$ be a compact convex graph and define \[ T(\Gamma) = \inf \left\{ t: \dim_{\mathrm H}(E)>t \Longrightarrow |E+\Gamma|>0 \text{ for every compact }E\subset\mathbb R^2 \right\}. \] For a graph over an interval of positive length the smallest possible value is $T(\Gamma)=1$. We ask whether this optimal conclusion can hold when pointwise Fourier decay of arclength is unavailable. The answer is yes, even for strictly convex curves. We use the Fourier transform convention $\widehat\nu(\xi)=\int e^{-2\pi i x\cdot\xi}\,d\nu(x)$. We construct a strictly convex Lipschitz graph $\Gamma$ with $T(\Gamma)=1$ such that, for every nontrivial subarc $\Gamma_0$ and every $\alpha>0$, \[ \limsup_{|\xi|\to\infty} |\xi|^\alpha \left| \widehat{H^1|_{\Gamma_0}}(\xi) \right|= \infty. \] We also give a convex example for which arclength on every nontrivial subarc fails even to be a Rajchman measure. The geometric mechanism behind these examples is a positive curved trace: if $\Gamma$ contains a positive-length subset of a $C^2$ curve whose curvature is bounded away from zero, then $|E+\Gamma|>0$ whenever $\dim_{\mathrm H}(E)>1$. For a nondegenerate graph this gives $T(\Gamma)=1$. For convex graphs it implies, in particular, that $T(\Gamma)=1$ whenever the curvature measure has a nonzero absolutely continuous part. The positive-measure proofs are in physical space and use translated-tube intersections and elementary facts about convex functions. The same overlap estimates give Mattila-type lower bounds for the average lengths of the associated curve projections of neighborhoods under the positive curved-trace hypothesis. We also prove a dimension-one endpoint result for sets with a positive-length rectifiable part and formulate the main remaining question: whether every strictly convex Lipschitz graph has the optimal threshold $T(\Gamma)=1$.

We show that the literal arbitrary-packet form of the refined-decoupling estimate printed in the Guth–Iosevich-Ou-Wang proof of the planar pinned Falconer theorem is false, even after its tube dimensions are normalized. An explicit collar construction gives a fixed-power counterexample: all packets are active on one square while none of their smaller labelled tubes meets that square. We then give a non circular repair of the original proof route. The analytic input is an enlargement-stable arbitrary-packet theorem in which a packet is concentrated on an $a$-dilate and multiplicity is counted with a strictly larger $b$-dilate. We prove this theorem directly from weighted $\ell^2$ decoupling by an induction that tracks the dilation margin through parabolic rescaling. The corrected theorem applies directly to the original Falconer parent packets after an exact frequency truncation and an absolute small-packet cutoff; no parent-to-canonical decomposition is needed. We then rebuild the good-tube incidence estimate, retain the neighborhood forced by local constancy, and supply a uniform regularization and limiting argument. The principal frequency exponent remains $-(\alpha+1)/3$, so the energy argument closes exactly for $\alpha>5/4$. The pinned theorem itself is not contradicted and is also known through later microlocal methods. A later canonical wave-packet treatment of refined decoupling overlaps with the activity-tube viewpoint but not with the counterexample or the repaired Falconer proof chain.

In this paper, we provide a general Sunada–Pesce–Sutton-type method that produces pairs of isospectral metric measure spaces. This construction relies on a representation theoretic assumption, and we give examples of spaces satisfying this condition, such as RCD spaces. As an application, we construct a pair of simply connected isospectral, non-isometric RCD non-Alexandrov spaces and a pair of Alexandrov non-orbifold spaces.

We study the combinatorial (red-blue) discrepancy of finite point sets with respect to hyperplanes and, more generally, bounded-complexity affine algebraic sets. We prove that every $n$-point set in a real Euclidean space admits a red-blue coloring for which every affine algebraic set of dimension at most $D$ and degree at most $k$ has discrepancy at most $n^{\frac12-\frac{1}{2(D+1)}-\varepsilon}$ for some $\varepsilon=\varepsilon(D,k)>0$. This gives a polynomial improvement over the straightforward VC-dimension bound $\tilde O(n^{\frac12-\frac{1}{2(D+1)}})$. In the opposite direction, we construct $n$-point sets in $\mathbb R^d$ whose discrepancy with respect to hyperplanes is $\tilde\Omega(n^{\frac12-\frac{1}{d+1}}),$ extending the point-line discrepancy lower bound of Chazelle and Lvov. We present further applications of our methods in communication complexity, concerning separation between randomized communication cost and deterministic communication cost with access to equality oracle.

We study which probability measures on $[0,\infty)$ can occur as the distribution of the distance between two independent points sampled from a complete separable metric space equipped with a Borel probability measure. We prove that an atomless Borel probability measure can be realised in this way if and only if its support contains the origin. This settles the conjecture of Aldous, Blanc, and Curien for absolutely continuous laws and also covers atomless measures that are singular with respect to Lebesgue measure. Moreover, for every $L>1$, we show that the realising metric may be chosen $L$-bi-Lipschitz equivalent to an ultrametric. The proof constructs the space from compact components whose mutual distances produce prescribed parts of the measure, while distances within the components are assigned to smaller scales.

Let $d\geq 2$, $0<\alpha<1$, and let $\Gamma_k$ be the successive arrival times of a standard Poisson process on $(0,\infty)$. Given independent uniform directions $\varepsilon_k\in S^{d-1}$, independent of $(\Gamma_k)$, we consider the random countable stable zonotope $Z_\alpha=\bigoplus_{k=1}^{\infty}\Gamma_k^{-1/\alpha}[0,\varepsilon_k]$. For its set of extreme points $\operatorname{ext} Z_\alpha$, we prove that almost surely $\dim_H \operatorname{ext} Z_\alpha=(d-1)\alpha$, and that the critical Hausdorff measure $\mathcal H^{(d-1)\alpha}(\operatorname{ext} Z_\alpha)$ is almost surely finite. The lower bound follows from the tangential non-degeneracy of the stable increments of the parametrizing field and Frostman's energy criterion. For the upper bound we construct an adaptive covering: at each scale the Poisson jumps are split into large and small ones, the large jumps determine a finite hyperplane arrangement, and the sum of the small jumps controls the diameters of the images of its cells.

For a real binary form with no real roots, Julia's zero and the hyperbolic zero are two SL(2,R)-equivariant points in the upper half-plane. We show that they admit closely related equilibrium characterizations, with radial weights given respectively by the hyperbolic tangent and hyperbolic sine of the distances to the roots. This common framework gives geometric criteria for coincidence of the two zero maps. They always agree for binary quartics; for binary sextics they agree exactly when the three upper-half-plane roots form an equilateral hyperbolic triangle, or are collinear with one root the hyperbolic midpoint of the other two. We also obtain a corresponding result for collinear binary octics. The two equilibrium laws further reveal a sharp difference in the influence of distant roots. A strict majority of roots confined to a compact set keeps Julia's zero in a compact set, and the threshold one-half is optimal. In contrast, an escaping minority can force the hyperbolic zero to escape at linear scale. We also illustrate the computational advantages of the explicit formula for the hyperbolic zero.

We study $\ell_0$ isoperimetry for a convex body $K\subset \mathbb{R}^n$, $n\ge2$. For a Borel set $S\subset K$, let $\partial_0^K S$ be the set of points in $K \setminus S$ that can be reached from $S$ by changing at most one coordinate (i.e. the $\ell_0$ boundary of $S$). Suppose that, for some unconditional convex body $Q \subset \mathbb{R}^n$, numbers $r,R>0$, and possibly different centers $x_0,y_0$, \[ x_0+rQ \subset K\subset y_0+RQ. \] Writing $s=\text{vol}(S)/\text{vol}(K)$, we prove that whenever $0<s \le 1/2$, \[ \frac{\text{vol}(\partial_0^K S)}{\text{vol}(S)} \ge \frac{cr}{nR} \min\left\{1,\frac{\log(e/s)}{n}\right\}, \] where $c > 0$ is an absolute constant. Consequently, the associated $\ell_0$-isoperimetric coefficient is at least $cr/(n^2R)$. Previous direct lower bounds were only known for $\ell_2$ and $\ell_\infty$ regularity whereas our lower bound holds directly for any $Q$-regularity, where $Q$ is an unconditional convex body. Compared to $\ell_2$ and $\ell_\infty$ regularity, our lower bound result improves upon the previously best known lower bounds, for any $s$, by a factor of $n$. As an application of our result, we give improved mixing time bounds for the Coordinate Hit and Run walk (CHAR). Our proof of the lower bound is based on a modification of the method of canonical paths applied to a continuous Hamming graph over our convex body. Our construction of canonical paths can be viewed as a suitable coordinate discretization of certain mass transport maps from $S$ to $S^c$. We also give complementary upper-bounds for any $Q$-regularity, with an overall factor of $n$ gap between the two.

We introduce the affine reflection transform (ART) of an oval $C$ relative to an interior point $p$ by applying the parallel-tangent involution to chords through $p$ and taking the envelope. A dual-projective approach gives a coordinate-free singularity criterion and proves that every generic non-degenerate ART has an odd number of ordinary cusps, at least three. Its relation to the centre symmetry set and the Wigner caustic yields further cusp bounds. We define a discrete ART for convex polygons with parallel opposite sides. Its combinatorial cells admit explicit rational parametrisations. Non-degenerate cells are arcs of ellipses or hyperbolas, and never parabolas. Under refining tangent approximations, the discrete transform converges in the Hausdorff metric to the smooth one. The oriented area of the smooth and polygonal transforms is non-positive and admits a Sobolev-type interpretation. Maximising its absolute value produces an affine-invariant asymmetry measure and a set-valued affine centre. We establish boundary degeneration and vanishing results, characterise central symmetry, and show that the maximising locus need not lie on the centre symmetry set or Wigner caustic. Finally, we prove universal upper bounds for the normalised energy, strengthen them for ovals of constant width.

We prove the spherical Hadwiger classification in every dimension. For $n\geq1$, every continuous $SO(n+1)$-invariant valuation on the space of all closed spherical convex sets in $\mathbb{S}^n$ can be written uniquely as a linear combination of the spherical intrinsic volumes $V_0,\ldots,V_n$. The same classification holds when the domain is restricted to sets contained in an open hemisphere. The proof is inductive: it reduces the problem to a vanishing statement for simple valuations and uses a continuous alternating cocycle on tuples of spherical points together with a signed coning transform. Through the cone–sphere correspondence, this yields, for $d\geq2$, the corresponding classification of continuous $SO(d)$-invariant conic valuations on all closed convex cones in $\mathbb{R}^d$, without imposing normalization at the zero cone. In particular, $SO$-invariance implies $O$-invariance in both settings.

We prove bounds of order $n^{n/2}e^{O(n)}$ for the expected number of facets of high-dimensional random polytopes. First, let $\mu$ be a non-degenerate compactly supported even probability measure on $\R$ satisfying $\mu([x^\ast-s,x^\ast])\asymp s^\kappa$ near its right endpoint $x^\ast$. For every sufficiently small fixed $\alpha>0$, the convex hull of $N=\lfloor e^{\alpha n}\rfloor$ independent points with law $\mu^{\otimes n}$ has at least $n^{n/2}e^{-C_{\mu,\alpha}n}$ expected facets; this includes all symmetric finite-alphabet distributions. For every full-dimensional log-concave probability measure on $\R^n$, we prove that there exist $T\in[n,2n]$ and $N=\lceil e^Tn^{3/2}\rceil$ for which \[ n^{n/2}e^{-Cn} \leq \mathbb E f_{n-1}(P_N) \leq n^{n/2}e^{Cn}. \] Thus the scale $n^{n/2}$, up to exponential factors, is universal for log-concave measures in this high-dimensional exponential regime. Finally, we construct a symmetric isotropic full-support non-log-concave counterexample with only $(1+o(1))2^n$ expected facets.

We study optimization problems for separable functionals of the Euclidean or spherical lengths of dual edge pairs in finite extremal unit-diameter configurations in three dimensions. For a fixed diameter graph, these problems lead to nonconvex constrained optimization of the vertex coordinates. We first prove that a convergent sequence of extremal configurations retains an extremal geometric core after coincident points are merged and vertices incident to at most one diameter are removed. A spherical Crofton argument then gives a sharp lower bound for additive concave functionals, attained by the regular tetrahedron. We also analyze the effect of inserting or deleting dangling vertices. For the sum of products of spherical dual-edge lengths, we obtain an exact supremal reformulation of the three-dimensional Blaschke–Lebesgue area problem. The numerical study uses all 10,644 available extremal configurations with at most 16 vertices. We combine direct evaluation with gradient-based local optimization on each fixed graph and reconstruct the intrinsic diameter graph and its dual pairs after vertex collisions. For every supplied graph, the selected verified endpoint contains a regular tetrahedron. These computations do not certify the global maximum for any fixed graph, but they motivate structural conjectures connecting tetrahedral containment with the Blaschke–Lebesgue problem.

Reformulated in terms of the asymptotic rank, a conjecture by Gromov predicts a linear isoperimetric filling inequality in all dimensions greater than or equal to the asymptotic rank of a Hadamard space, in contrast to the Euclidean-type nonlinear behavior below this threshold. We prove the predicted linear inequality for Hadamard spaces with finite asymptotic Nagata dimension and finite asymptotic rank. More precisely, every integral cycle of dimension at or above the asymptotic rank admits a filling whose mass is bounded linearly in the mass of the cycle. Our proof is based on a new self-improvement mechanism for Wenger's sub-Euclidean growth theorem. This approach upgrades the asymptotic rank-one result by Wenger and the recent asymptotic rank-two result by Lang, Stadler, and Urech from exponents arbitrarily close to one to the optimal linear exponent. Moreover, the result extends to arbitrary finite asymptotic ranks.

We show that every continuous and isometry invariant valuation on spherical polytopes is a linear combination of the spherical intrinsic volumes. The proof relies on a weak differentiability property satisfied by valuations on polytopes in $\mathbb{R}^n$ with a natural smoothness property with respect to the action of the affine group. This enables us to transfer several results established by Alesker for quasi-smooth valuations to the polytopal setting, and to reduce the problem to the translation invariant case of measurable valuations on polytopes.

We present a discrete Morse theoretic approach to proving high connectivity or contractibility of a Vietoris-Rips complex using distance to a fixed point as an initial measurement. In particular we focus on the case of a finitely generated group using word length. As a proof-of-concept application we prove that $\mathcal{VR}_2(A_\Gamma)$ is contractible for $A_\Gamma$ a right-angled Artin group on a triangle-free graph $\Gamma$. We also prove an interesting sufficient condition for a group to be finitely presented that only requires checking connectivity of certain finite complexes.

The set of $\tau$-approximable numbers, $\mathcal W(\tau)$, has genuinely fractional Hausdorff dimension, whereas the set of inhomogeneously badly approximable numbers, $\Bad^\gamma$, has full Hausdorff dimension. We determine the Hausdorff dimension of their intersection by introducing the $\Psi$-rapid game, a scale-sensitive refinement of the rapid game of Hatefi and Simmons (preprint 2024). For every approximation function $\psi$, we prove that $\mathcal W(\psi)\cap\Bad^\gamma$ is strong $\Psi$-rapid winning for a natural gauge $\Psi$ determined by $\psi$. Unlike Schmidt-type games, whose winning property always implies full Hausdorff dimension, the $\Psi$-rapid game is calibrated to a prescribed Diophantine scale, so that the resulting dimension bound depends explicitly on the decay of $\Psi$. In particular, for $\psi(q)=q^{-\tau}, \tau\ge1,$ we recover the exact Jarník–Besicovitch dimension, that is, $$ \HD\bigl(\mathcal W(\tau)\cap\Bad^\gamma\bigr)=\frac{2}{\tau+1}.$$

We prove thirteen results concerning three aspects of magnitude theory. The first four concern small-scale magnitude and microscopic weightings. We identify the derivative of the magnitude function at the origin with maximal energy, allowing the value $+\infty$, and construct finite spaces for which small-scale magnitude diverges or for which the one-point property and finite concentration do not produce a microscopic weighting. A fifth result disproves the conjectured monotonicity of magnitude by giving a fourteen-point metric space of strict negative type whose magnitude function decreases. The remaining eight results concern continuity and dimension. Magnitude is Hausdorff-continuous at finite subsets of finite-dimensional positive definite normed spaces, but need not be continuous on compact subsets, even under convexity, geodesicity, bounded tractability, or simultaneous Hausdorff and Lipschitz convergence. The same constructions disprove an $\ell_1$-intrinsic-volume formula and two tractability conjectures. Finally, we construct a countable compact space of negative type whose Minkowski dimension is zero and whose lower and upper magnitude dimensions are both infinite.

A point lies in the kernel of a polygon if it can see the entire polygon. Thus the kernel measures how much of the polygon is available to a single guard, while the convex hull measures how far the polygon is from being convex. We prove that these two losses are linked by a sharp factor of two: the area lost between a polygon and its kernel dominates twice the kernel-weighted area missing from the polygon's convex hull. In terms of Sibley's guard-point ratio $G$ and exterior ratio $E$, which we denote by $A$, the result is $G \leq A/(2-A)$, which improves Nakano's inequality $G \leq A$ whenever $A < 1$. The proof passes through a convex-body cap union. From a compact convex body $K$ and finitely many points whose convex hull contains it, we join every point to $K$ and take the union $U$ of the resulting caps. Cyclically sorting the directed boundary edges of a polygonal $U$ produces a convex companion $H$. A boundary-reversal argument gives $|H| + |U| \geq 2|\mathrm{conv}\, U|$, while a support-function identity and Minkowski's mixed-area inequality give $|U|^2 \geq |K||H|$. Inner polygonal approximation handles every positive-area compact convex $K$, while a separate null-area branch covers points, segments, and all other lower-dimensional cases. Both main theorems have machine-checked Lean 4 proofs whose final statements were audited against the informal statements after kernel checking.

In 1893, James Joseph Sylvester posed the following problem: given n points in the plane, not all collinear, must there be a line determined by two of the points that does not pass through any of the other points? In 1940, Eberhard Melchior studied the equivalent dual problem in the projective plane: given a set of n lines in the (real) projective plane, not all passing through a common point, must there be a point where exactly two of the lines intersect? Such a point of intersection is called an *ordinary point*. Via a clever double-counting argument, Melchior found that in fact there must be at least three such points. Given the many simple "visual" proofs of what is today known as the Sylvester-Gallai Theorem — the theorem that states there must be at least one ordinary point — a natural question is whether there is a simple visual proof that recovers all three of Melchior's ordinary points. This paper provides such a proof.