Rotation-invariant valuations on the sphere are classified
Every continuous one is a linear combination of spherical intrinsic volumes, settling the spherical case.
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Metric Geometry
Euclidean, hyperbolic, discrete, convex, coarse geometry, comparisons in Riemannian geometry, symmetric spaces
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Round disk demands the greatest distortion in map projections
Every equal-area convex region fits in a plane map with distortion at most rho/sin rho.
Exact third-order area constant for largest even small polygons
A single algebraic rate governs the alternating, exponentially damped edge profile of maximum-area even polygons.
· “Boundary Layers and Sharp Asymptotics for Maximum-Area Small Polygons”
Submetries tie three metric geometries into one chain
Geodesic and curvature results for three spaces all reduce to one chain of maps.
Log-submodularity of zonoid volumes fails in every dimension n≥4
Sharp threshold at degree or codegree 3 gives counterexamples that also sink Courtade's conjecture for zonoids in n≥3.
· “Volume and Projection Inequalities I: Zonoids and Courtade's Conjecture”
Construction finishes the 196560 Leech-lattice conjecture
A closed ray of Schwartz functions yields the 196560 ratio through Poisson summation.
· “The 196560 auxiliary-function conjecture for the Leech lattice”
Boundary topology fixes the quasi-isometry class of Kleinian groups
Kleinian groups are quasi-isometric exactly when Bowditch boundaries agree on carpets and cut-point orders.
Quasi-planar graphs are built from planar graphs by two moves
Every graph that looks planar from far away has a finite combinatorial recipe from planar graphs.
Centroid alignment beats 2/(n+1)^n of optimal overlap
The gap left open since 1996 is closed: 4/9 is the exact worst case in the plane.
Identification capacity capped by entanglement-assisted rate
Every quantum channel's ID rate is capped by its entanglement-assisted capacity; low-noise channels meet it.
· “The entanglement-assisted transmission capacity is a strong converse bound for identification”
Polyhedron immerses flat Klein bottle in 3-space
Zero defect at every vertex and embedded stars produce the first explicit piecewise isometric example
Coarea formula holds for maps from Heisenberg group to R^{2n}
A new integral for the area of half-Holder curves makes this work even for the simplest vector-valued case.
· “Area of H\"older curves and coarea formula on the Heisenberg group”
This paper proves the spherical Hadwiger classification in every dimension: continuous…
Every continuous SO(n+1)-invariant valuation on closed spherical convex sets in S^n is a unique linear combination of the spherical…
First exact n^2 log n Dehn function for nilpotent groups
2-step nilpotent Lie groups and their lattices fill loops with area n^2 log n, ending the power-only pattern.
Every dimension has an infinite rational-distance set
One Chebyshev construction yields rational chords and full affine independence in every dimension.
· “Infinite rational distance sets in affine general position: constructions in every dimension”
Distance metric decides when triangle badness ranks match
Two of three triangle-quality measures stay consistent across most datasets; the third measures something else.
· “An Approach to Study the Structural Consistency of Triangle Badness Functions and Distance Metrics”
Graph Laplacian spectra converge without smoothness assumptions
A coarse Poincaré inequality, not a manifold, is enough: error is 1/√(n v(h)) for simple eigenvalues.
· “Spectral stability of empirical metric-measure Laplacians”
The paper proves that the volume of the largest centered ellipsoid fitting inside a…
For any pseudoconvex family of domains, the negative log of the maximal inscribed ellipsoid volume is plurisubharmonic, and no equivariant…
Claw–tripod count proves Steiner-tree uniqueness
Equal numbers of claws and tripods in a plane graph force codirected minimal trees to coincide.
Shift-symmetric curves cap distinct-volume sets at sqrt(2an)
New point and hyperplane constructions pin down how large a subset must be before two simplices share a volume.
Seven Circles Theorem extends to all closed six-circle chains
Two helper circles replace the common circle; the six tangency points become circles, and opposite center-lines still concur.
· “When Points Become Circles: The Seven Circles Theorem for Arbitrary Closed Six-Circle Chains”
Unique dual L_p John ellipsoids exist for general measures
A measure-weighted minimization has one optimal ellipsoid, extending John and Loewner ellipsoids to Gaussian measures.
ℓ_p projection volumes move quadratically away from extremes
Near any extremal hyperplane, the ℓ_p-ball projection ratio deviates by at least c_p times squared distance.
· “Stability of extremal hyperplane projections of balls in ell_p^n(mathbb{R})”
Short proof fixes circle-to-odd-sphere distance at πk/(2k+1)
The upper bound matches the known lower bound, so the Gromov–Hausdorff distance is exact for every odd sphere.
· “Tight upper bound on d_(GH)(S¹,S^(2k+1)): GPT's short proof”
Squares now fill cubes at the sharp 2/3-Hölder exponent
The paper settles the attainability half of Arnold's 1988 problem with explicit Hölder-continuous space-filling surfaces.
· “Space-filling surfaces: sharp H\"older continuous parameterizations from squares to cubes”
Macroscopic curvature forces 1-width bound in all dimensions
Complete manifolds with small balls and fillable cycles have width at most 24(b+1)t_{n-2}.
Boundary mass governs when disk iterates hit the sharp growth rate
A logarithmic integral of the Herglotz measure decides whether hyperbolic distance grows with bounded error.
· “Characterizations of extremal hyperbolic rates via Herglotz measures and Koenigs linearization”
Two commuting maps with bounded fibers give hyperfinite relations
New proof: any two commuting bounded-to-one Borel maps generate an equivalence relation that is hyperfinite.
· “Hyperfiniteness of bounded-to-one actions of commutative monoids”
Shoreline search costs at least 12.5937 times optimal
The unconditional bound nearly doubles the old 6.3972 and applies to any deterministic path.
Distortion of cluster-graph edit distance pinned at Θ(n^{1/4})
Explicit O(n)-time map hits optimal Euclidean distortion; 2-approximation and hardness dichotomy follow.
· “Cluster-Graph Edit Distance: Optimal Explicit Embeddings, Metric Proxies, and Complexity”
Planar lamplighter rejects all Hilbert embeddings
For finite torus quotients the Euclidean distortion grows at least as sqrt(log n), settling an L1 embedding question.
Curvature of interpolated gene networks recovers fly stage order
No intermediate samples needed: Gromov-Wasserstein geodesics plus Ollivier-Ricci curvature rank real stages correctly.
· “A Unified Geometric Framework for Developmental Analysis of Spatial Transcriptomic Data”
Explicit map transports polar-band masses optimally on the sphere
The same meridian-sliding formula is optimal for every p-cost between antipodal spherical bands.
Contours govern persistence-module distances
A bijection ties simple norms to contours; continuous contours yield compact families.
Octahedron matches cube's Nieuwland number exactly
A polar-duality proof settles the conjecture that the octahedron's passage constant is 3√2/4.
4D zonoid volumes obey a new projection inequality
The proof reduces the inequality to a hypercube polynomial and certifies it by sums of squares.
· “On the Log-submodularity for zonoids: from Mixed Volume inequalities to the Hypercube”
Mean-distance conjecture: true in plane
A 2019 conjecture on random-point distances is settled for symmetric bodies, with counterexamples in every dimension 3 and up.
· “Mean distance between points inside and on the boundary of a convex body”
Equal-distance planes are finite in every dimension
The line case was known; a double-exponential bound now covers every dimension pair.
Mean distance is less inside a convex body than on its boundary
Settles the 2019 conjecture: strict inequality holds for every planar convex body.
· “The Zaporozhets-Tarasov Inequality for an Arbitrary Planar Convex Body”
Lottery complexity is distance from the nearest sure outcome
A new axiom system recovers the exact metric behind perceived complexity, so 'twice as complex' becomes meaningful.
3D point sets force N^{2/3-o(1)} distinct distances
The integer lattice is asymptotically optimal for the 3D Erdős distinct distances problem, up to a subpolynomial factor.
A single quadratic twist of the measure proves the slicing theorem
No stochastic localization needed: a covariance-survival principle and Lp-centroid bodies bound the isotropic constant universally.
Nonnegative curvature forces groups to be virtually abelian
In polynomial-growth groups, only virtually abelian groups admit a nonnegative-curvature Cayley graph.
· “Ollivier--Ricci Curvature on Groups of Polynomial Growth”
Ricci flow and its Wasserstein cousin agree only to first order
The metric moves even where Ricci flow stands still, with curvature setting the size of the t² gap.
· “Second-Order Departure of the Gigli--Mantegazza Flow from Ricci Flow”
58 points break Kusner's equilateral-set conjecture
An explicit set in ℓ5^56 exceeds the n+1 bound for the first time for any finite p≥2.
· “A counterexample to Kusner's conjecture on equilateral sets”
Ellipses are the only planar shapes with the Hausdorff bound
A complete theorem: the squared subadditive bound holds exactly for centered ellipses, with all equality cases.
· “Norm rigidity and equality cases for the Dyn--Farkhi inequality”
Symplectic self-polar bodies all deform to one fixed body
A rotating isotropic subspace gives a continuous path between any two such bodies, making the whole space contractible.
· “Contractibility of space of symplectically self-polar convex bodies”
The gap between adjacent spheres is exactly half a simplex edge
One explicit pairing of points matches the topological lower bound, settling all previously open cases n ≥ 4.
· “The Gromov-Hausdorff Distance Between Consecutive Spheres”
Lipschitz maps into any Banach space have compact tangent sets
Preiss's phenomenon now covers arbitrary Banach targets and pmGH tangents under a porous-set condition.
Approximate high-mass maps force the target into pyramid limits
Nearly every point's worth of directed data carries a target representation into every subsequential weak limit.
· “Extensions of One-Sided Box Geometry and Pyramid Invariants to gd-Sets and qm-Spaces”
Exact triangle-cover threshold found for every unit rhombus
Two exact formulas give the least total area of parallel equilateral triangles that guarantees coverage; less can fail.
First counterexample: lamplighter word-length triples are not metric
For every d≥2 and every generating set, the diagonal operator on the lamplighter group fails to metrize the state space.
· “Word-Length Spectral Triples of (mathbb{Z}/2mathbb{Z})wrmathbb{F}_(d) Are Not Metric”
Two-pencil circle surfaces fully classified
A quaternionic factorization test sorts them into sums, products, cubics, and quadrics.
· “Bivariate quaternionic factorizations and surfaces that decompose into two circles”
Uniform discreteness and Pisot scaling are equivalent for Delone sets
A scaling factor below 2 forces Pisot numbers; Pisot factors force patterns to be Meyer sets.
Spherical designs make near-round polytopes at rate t^{-1/2}
Every t-design's equal-area Minkowski polytope sits within O(t^{-1/2}) of the unit ball, with no separation assumptions.
· “Minkowski Polytopes of Spherical Designs: High-Order Isotropy and Quantitative Sphericity”
Submodular set functions are convex covering radii
New embeddings show submodular and XOS functions measure how much a convex kernel must stretch to cover a set.
· “Submodular and strongly submodular functions and diversities”
Same lattice-point counts for every shift force sets to agree
A short Fourier argument settles open conjectures on reconstructing convex bodies and polytopes from Ehrhart-like data.
· “A Fourier-analytic Uniqueness Theorem for Lattice-point Enumerators”
Floating-body corners force polytopes into prisms
The rigidity result pins down what local minimizers of Mahler's volume product can look like.
· “A connection between the Mahler conjecture and floating bodies”
Moment comparison nails Sudakov constant at sqrt(log n)
Same argument cuts the weak-strong moment coefficient to n^{1/4} polylog.
· “Moment comparisons, Sudakov inequalities and entropy of centroid bodies”
Every admissible curve-projection family has a finite witness
New hypotheses let Federer's projection theorem pass through critical sets, settling the rectifiable endpoint.
· “Finite Good Witnesses for Generalized Curve Projections at the Rectifiable Endpoint”
Sharp Rogers–Shephard inequality holds for all α-concave functions
Equality forces simplex indicators except in dimension one, where monotone functions are extremal.
· “Integral inequalities for α-convolutions of α-concave functions”
Half a circle is the limit for ℓ1 embeddings
Closed circle subsets embed in ℓ1 exactly when a half-circle holds all but measure zero.
Every convex body has a bounded subgaussian body
Its volume ratio to the L2-centroid body is an absolute constant, for all dimensions.
· “Geometry of the subgaussian body of an isotropic convex body”
New curvature for complex-weighted graphs recovers the directed case
With k-th-root-of-unity phases, it reduces to a layered directed graph and finds communities undirected curvature misses.
Three normals must meet at every optimal chord
The same rule behind Philo's line locates every extremal normal chord between two curves.
· “Geometric optimization problems generated by plane curves”
One random Lipschitz function catches every jump in a metric space
Positive LCJ constant is equivalent to a random 1-Lipschitz function whose expected difference is proportional to distance.
Minimal étale groupoids are Følner or paradoxical
A new coarse-geometric notion unifies group amenability and metric amenability and yields invariant measures.
One monotonicity law unifies Cheeger
Graph limits, hypergraphs, manifolds and metric measure spaces obey the same inequality.
Metric graph length capped by diameter
The bound is tight, and equality holds exactly for the Medusa graphs.
Near-equality in the (B)-theorem forces eigenvalues near 0 or 1
Dimension-free estimate: every covariance eigenvalue is O(δ) or 1−O(δ), independent of dimension.
Second-largest Kähler volume found: quadric or P^1×P^{n-1}
Ricci-positive Kähler manifolds beyond projective space cap at 2 n^n/(n+1)^n of Fubini–Study volume, and the bound is attained.
· “The sharp volume gap for K\"ahler manifolds with positive Ricci curvature”
A 4-D zonotope refutes log-submodular volumes
Explicit generators give 14×56 > 30×26, so the conjecture fails for all zonoids in dimensions ≥4.
New infinite family of polytopes with unrealizable central symmetry
Bier spheres give cs-neighborly examples whose combinatorial symmetry no convex realization can respect.
· “Polytopal Bier spheres and nonrealizable central symmetries”
Isotropic bodies get M M* ≤ C log² n
Deterministic dyadic curvature arguments prove both bounds without stochastic methods.
· “Geometric Bounds for the Mean Gauge and the Mean Width in Isotropic Position”
Basic tropical polynomials generate all r-symmetric rational functions
Explicit generators of degree at most n replace a bound using the first n! primes.
· “The basic tropical polynomials generate the semifield of r-symmetric tropical rational functions”
Five unit disks have minimum hull perimeter 10+2π
First solution beyond four disks: three contact classes attain the bound, two flex, one is rigid.
First d width moments recover any box's side lengths
Spectral endpoints obey the same identity, and face-scale corrections keep shape memory beyond universality.
Slab counting proves Busemann's intersection inequality
New empirical dual volumes make the classical inequality follow from counting random points in Gaussian slabs
One sequence can carry a fractal's full dimension
Upper box, Assouad, intermediate and profile dimensions can all be witnessed by one convergent sequence.
Radon numbers can grow quadratically in a simple construction
In a finite grid, the d-th Radon number reaches Θ(d log² d) while the base case stays Θ(log d).