Geometry & Topology
15 results · 30 papers · 13 with a Lean formalization
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 corresponding sharp functional Mahler inequalities also hold. For n ≥ 2, every symmetric polar product \(K\times K^\circ\) in dimension \(2n\) has Gromov width 4.
3 papersLeanreasoningProves 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 contrast, products of simplices exceed Brannen's proposed simplex maximum for normalized projection-body volume by an exponential factor in every sufficiently large dimension.
2 papersLeanResolves 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 distortion for planar graphs, and distortion depending only on treewidth for bounded-treewidth graphs. The corresponding multicommodity flow–cut gaps are uniformly bounded.
2 papersLeanProves 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 density one. It also minimizes unit-background renormalized Riesz energies for \(0\lt s\lt 2\) and Coulomb energy, resolving Sandier–Serfaty and the two-dimensional Brauchart–Hardin–Saff conjecture on the linear term of optimal spherical logarithmic energy.
4 papersLeanProves 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 the additive Lp Brunn–Minkowski inequality for full-dimensional origin-symmetric convex bodies throughout \(0\lt p\lt 1\).
1 paperLeanDetermines 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 \(Cn\log n\); centrally symmetric examples in every sufficiently large dimension require at least \(cn\log n\) even without the lattice restriction, for absolute \(c,C\gt 0\).
2 papersLeanProves 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 parameter.
1 paperFor every fixed \(1\lt p\lt \infty\) and distortion D > 1, every n-point subset of real Lp embeds into \(\ell_p^d\) with distortion at most D and dimension \(d=n^{o(1)}\), answering Naor's sublinear-dimension question for p ≠ 2. In contrast, exact embeddings require worst-case dimension \(\Theta(n^2)\) when p ≠ 2.
1 paperLeanDisproves 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 coefficients used. This also disproves the generalized Lax conjecture that every hyperbolicity cone is spectrahedral.
3 papersLeanProves 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 bound. Combined with the separate Unique Games theorem, this proves NP-hardness of improving the loss factor \((8\pi/9)(1-1/k)\) for identity-target kernel clustering with fixed k ≥ 3 on rational centered positive semidefinite inputs.
1 paperLeanProves 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 bound on unsigned partial sums. A matching lower bound gives the optimal order \(S_2(d)=\Theta(\sqrt d)\).
1 paperLeanEvery 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 Lang–Plaut problem negatively, even for compact subsets of Hilbert space.
1 paperLeanDetermines 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 constants are uniform over all finite alphabets with at least two symbols, even when the alphabet grows with d; binary words already force the lower bound.
3 papersLeanCovers 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 half-area cylinder-covering bound and the stronger directionwise normalized conjecture in dimension three.
4 papersLeanProves 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 probability densities, with equality precisely for invertible affine images of products of one-sided exponential laws.
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