Convex and metric geometry

Convex and metric geometry15

• 087Sep 22, 2026 Explained

  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 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.

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• 088Sep 24, 2026 Explained

  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 contrast, products of simplices exceed Brannen's proposed simplex maximum for normalized projection-body volume by an exponential factor in every sufficiently large dimension.

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• 089Sep 23, 2026 Explained

  Bounded-distortion L₁ 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 distortion for planar graphs, and distortion depending only on treewidth for bounded-treewidth graphs. The corresponding multicommodity flow–cut gaps are uniformly bounded.

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• 090Sep 23, 2026 Explained

  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 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.

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• 091Sep 23, 2026 Explained

  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 the additive L_p Brunn–Minkowski inequality for full-dimensional origin-symmetric convex bodies throughout $0\lt p\lt 1$.

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• 092Sep 23, 2026 Explained

  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 $Cn\log n$; centrally symmetric examples in every sufficiently large dimension require at least $cn\log n$ even without the lattice restriction, for absolute $c,C\gt 0$.

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• 093Sep 23, 2026 Explained

  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 parameter.

  Claimed, not checked

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• 094Sep 23, 2026 Explained

  Subpolynomial dimension reduction in L_p

  For every fixed $1\lt p\lt \infty$ and distortion D > 1, every n-point subset of real L_p 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.

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• 095Sep 24, 2026 Explained

  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 coefficients used. This also disproves the generalized Lax conjecture that every hyperbolicity cone is spectrahedral.

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• 096Sep 24, 2026 Explained

  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 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.

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• 097Sep 24, 2026 Explained

  The Euclidean Steinitz–Bergström bound

  Proves that any finite sequence in the Euclidean unit ball of ℝ^d 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 $S_2(d)=\Theta(\sqrt d)$.

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• 098Sep 25, 2026 Explained

  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 Lang–Plaut problem negatively, even for compact subsets of Hilbert space.

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• 099Sep 27, 2026 Explained

  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 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.

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• 100Sep 27, 2026 Explained

  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 half-area cylinder-covering bound and the stronger directionwise normalized conjecture in dimension three.

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• 101Oct 5, 2026 Explained

  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 probability densities, with equality precisely for invertible affine images of products of one-sided exponential laws.

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