Petty's Projection Conjecture, Log Brunn–Minkowski Conjecture and 8 More

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. (Lean)

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. (Lean)

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 < s < 2 and Coulomb energy, resolving Sandier–Serfaty and the two-dimensional Brauchart–Hardin–Saff conjecture on the linear term of optimal spherical logarithmic energy. (Lean)

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 Lp Brunn–Minkowski inequality for full-dimensional origin-symmetric convex bodies throughout 0 < p < 1. (Lean)

Determines the optimal worst-case covering density as Θ(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 > 0. (Lean)

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.

Proves that the sum of squared Gaussian first moments of any finite measurable partition is at most 9/(8π). In dimension at least two, three planar sectors of angle 2π/3, extended orthogonally, attain the bound. Combined with the separate Unique Games theorem, this proves NP-hardness of improving the loss factor (8π/9)(1-1/k) for identity-target kernel clustering with fixed k ≥ 3 on rational centered positive semidefinite inputs. (Lean)

Proves that any finite sequence in the Euclidean unit ball of ℝd admits signs keeping every partial sum within C√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 S₂(d)=Θ(√d). (Lean)

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. (Lean)

Determines the least distortion of embedding edit distance on words of length at most d into real ℓ1: it is exp(Θ(√(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. (Lean)