Metric Geometry
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Paper Hyperbolicity cones without semidefinite lifts
Disproves the Projected Lax conjecture: some hyperbolicity cones are not spectrahedral shadows. The examples admit no exact finite affine semidefinite lift, regardless of the number of auxiliary variables or the real…
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Paper The sharp simplex conjecture for isotropic constants
Proves that simplices uniquely maximize the isotropic constant among convex bodies in every dimension, resolving the strong isotropic constant conjecture. Also establishes the sharp entropy lower bound for log-concave…
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Paper The sharp exponential scale of edit-distance distortion
Determines the least distortion of embedding edit distance on words of length at most d into real ℓ1: it is \exp(\Theta(\sqrt{\log d\,\log\log d})). Insertions, deletions and substitutions have unit cost. The…
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Paper Cylinder coverings below the half-area bound
Covers the entire closed regular tetrahedron by finitely many cylinders with compact triangular perpendicular bases whose total area is less than half its smallest orthogonal projection area. This disproves Bang's…
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Paper Triangular-lattice optimality, long-range Riesz and Coulomb energies, and spherical logarithmic energy
Proves that the triangular lattice minimizes the lower limit of energy per particle for every nonnegative completely monotone potential of squared distance among locally finite planar configurations of centered-disk…
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Paper Compact counterexamples to bi-Lipschitz dimension reduction
Every infinite-dimensional real Banach space contains a compact doubling set that admits no bi-Lipschitz embedding into any finite-dimensional normed space. The doubling constant is universal. This answers the…
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Paper Sharp projection-body inequalities and a counterexample to simplex maximization
Proves Petty's projection-volume conjecture in the remaining dimensions n ≥ 4: ellipsoids uniquely minimize projection-body volume at fixed body volume. Also establishes the full Lutwak–Petty projection inequalities. In…
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Paper The Gaussian propeller conjecture in every dimension
Proves that the sum of squared Gaussian first moments of any finite measurable partition is at most 9/(8\pi). In dimension at least two, three planar sectors of angle 2\pi/3, extended orthogonally, attain the…
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Paper The Euclidean Steinitz–Bergström bound
Proves that any finite sequence in the Euclidean unit ball of ℝd admits signs keeping every partial sum within C\sqrt d, independently of length. Consequently every zero-sum family can be reordered with the same…
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Paper Logarithmic and L_{p} Brunn–Minkowski inequalities and the B-conjecture
Proves the logarithmic Brunn–Minkowski inequality for origin-symmetric convex bodies in every dimension, and the scalar-dilation B-conjecture for all even log-concave Radon measures. For Lebesgue volume it also proves…
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Paper Subpolynomial dimension reduction in L_{p}
For every fixed 1\lt p\lt \infty and distortion D > 1, every n-point subset of real Lp embeds into \ellp^d with distortion at most D and dimension d=n^{o(1)}, answering Naor's sublinear-dimension question…
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Paper Bounded-distortion L_{1} embeddings of planar and bounded-treewidth graphs
Resolves the planar and bounded-treewidth cases of the Gupta–Newman–Rabinovich–Sinclair conjecture. Shortest-path metrics of finite connected graphs with arbitrary positive edge lengths embed into real L1 with universal…
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Paper The optimal order of convex-body covering density
Determines the optimal worst-case covering density as \Theta(n\log n), for both lattice and unrestricted translative coverings. Every convex body in ℝn, n ≥ 2, admits a lattice covering of density at most $`Cn\log…
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Paper Dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures
Proves a dimension-free logarithmic Sobolev inequality for centered log-concave densities with uniformly subgaussian linear marginals, with constant bounded by a universal multiple of the squared linear subgaussian…
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Paper The Mahler conjectures, functional inequalities and polar-product symplectic width
Resolves the symmetric and nonsymmetric geometric Mahler conjectures in every dimension, with Hanner polytopes and simplices as the respective volume-product minimizers and all equality cases classified. The…
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