Convex and metric geometry

We resolve the symmetric Mahler conjecture positively, including its equality classification. Every origin-symmetric convex body in ℝⁿ has volume product at least \(4^n/n!\), with equality exactly for invertible linear images of Hanner polytopes.

We resolve the Mahler conjecture for general convex bodies positively. For every convex body \(K\subset\mathbb R^n\), n ≥ 1, with Santaló point \(s(K)\), \(|K|\,|(K-s(K))^\circ|\ge (n+1)^{n+1}/(n!)^2\), with equality exactly for simplices.

For every integer n ≥ 2 and every origin-symmetric convex body \(K\subset\mathbb R^n\), we prove that the Gromov width of \(\mathop{\mathrm{int}}\nolimits K\times\mathop{\mathrm{int}}\nolimits K^\circ\) is 4. We construct smooth symplectic embeddings of standard balls of every capacity \(0\lt c\lt 4\) into this polar product. Volume preservation then resolves the symmetric Mahler conjecture positively in every dimension.

We prove that ellipsoids uniquely minimize the volume of the projection body among convex bodies of fixed volume in every dimension at least four. This proves Petty's projection-volume conjecture in these dimensions.

The product of two ten-dimensional simplices has larger normalized projection-body volume than a twenty-dimensional simplex. This gives a counterexample to Brannen's proposed simplex maximum.

We prove that every finite connected planar graph with arbitrary positive real edge lengths embeds into real L₁ with a universal distortion bound. This resolves the planar embedding conjecture positively.

For every fixed treewidth bound, the shortest-path metrics of finite connected graphs with arbitrary positive real edge lengths embed into real L₁ with uniformly bounded distortion. This resolves the bounded-treewidth case of the Gupta–Newman–Rabinovich–Sinclair conjecture positively.

We prove that the density-one triangular lattice minimizes the lower energy per particle for every nonnegative completely monotone function of squared distance, among all locally finite planar configurations of centered disk density one. The comparison includes infinite energies. The proof constructs sharp Gaussian Fourier minorants using an atomic interpolation certificate.

We prove universal energy minimality for the triangular lattice in the plane. Among locally finite configurations of centered density one, it minimizes the lower limit of centered-ball energy averages for every nonnegative completely monotone function of squared distance, including when the energy is infinite. We also prove triangular minimality for planar logarithmic and Riesz renormalized energies, with \(0\lt s\lt 2\) in the Riesz case, and the corresponding jellium minima. The proof uses sharp Gaussian Fourier bounds, positive mixtures, and heat-kernel comparison.

We prove the Sandier–Serfaty conjecture: the triangular lattice of covolume one minimizes planar Coulomb renormalized energy over all admissible curl-free fields with a unit uniform background. Combined with Bétermin and Sandier's asymptotic formula, this also proves the Brauchart–Hardin–Saff conjecture for the linear term of optimal ordered-pair logarithmic energy on the unit two-sphere. The proof uses a direct Voronoi-cell comparison with rigorous interval arithmetic.

We resolve the planar Cohn–Elkies sharpness conjecture: the two-point Fourier bound attains the optimal circle-packing density \(\pi/(2\sqrt3)\). We construct a radial Schwartz certificate and prove its global sign conditions using rigorous interval arithmetic and analytic estimates. The same certificate recovers the classical uniqueness of the triangular packing among periodic equality cases.

We prove the logarithmic Brunn–Minkowski conjecture for arbitrary origin-symmetric convex bodies in every dimension. The theorem also gives the symmetric L_p Brunn–Minkowski inequality for every \(0\lt p\lt 1\). Combined with Saroglou's transfer theorem and a support-subspace reduction, it yields the logarithmic inequality for every even log-concave Radon measure and the scalar-dilation \((B)\)-conjecture.

For every sufficiently large dimension n, we construct a centrally symmetric convex body whose translative covering density exceeds \(c n\log n\), where c > 0 is absolute. This disproves the existence of a universal linear upper bound and matches the order of Rogers' upper bound.

Every convex body in ℝⁿ, n ≥ 2, admits a covering by translates along one full-rank lattice with density at most \(Cn\log n\), for an absolute constant C. No symmetry or boundary regularity is assumed.

We prove that every centered log-concave probability measure with a Lebesgue density on ℝⁿ and linear subgaussian parameter a satisfies \(\mathop{\mathrm{Ent}}\nolimits _\mu(f^2)\le Ca^2\int|Df|^2\,d\mu\) for compactly supported smooth f, with one universal constant C. This resolves positively the dimension-free logarithmic Sobolev conjecture for subgaussian log-concave measures.

For every fixed \(1\lt p\lt \infty\) and D > 1, every n-point subset of a real L_p space embeds into \(\ell_p^d\) with distortion at most D and subpolynomial dimension \(d=n^{o(1)}\). The target has the same exponent p, and the embedding need not be linear.

We prove that not every hyperbolicity cone is a spectrahedral shadow: some closed hyperbolicity cones admit no finite affine semidefinite lift, even with arbitrary real coefficients and any finite number of auxiliary variables. This disproves the Projected Lax Conjecture and hence the generalized Lax conjecture.

We construct an exact semidefinite lift of the explicit nonspectrahedral hyperbolicity cone in twenty-three variables defined in the companion paper. The lift is a homogeneous real symmetric pencil of size 100 with 307 auxiliary variables and represents the entire closed cone, including every point with singular X. Thus, although this cone has no semidefinite representation in its original coordinates, it admits one when auxiliary variables are allowed. The stated sizes are not claimed to be minimal.

We construct a homogeneous polynomial of degree 16 in 23 real variables whose hyperbolicity cone has no representation by a finite homogeneous real symmetric linear matrix inequality. This disproves the geometric Generalized Lax conjecture.

We prove the Gaussian propeller conjecture: for every finite measurable partition of a Euclidean space, the sum of the squared lengths of its Gaussian first moments is at most \(9/(8\pi)\). For dimension at least two and at least three cells, three planar sectors of angle \(2\pi/3\), extended by an orthogonal Euclidean factor, attain the bound.

Every prescribed-order finite sequence of vectors in the Euclidean unit ball of ℝ^d has one signing for which every signed prefix has norm at most \(C\sqrt d\), with C absolute and independent of the sequence length. Consequently, every indexed zero-sum family of unit-ball vectors admits an ordering with the same bound for its unsigned partial sums. This determines the Euclidean Steinitz constant up to absolute factors, \(S_2(d)=\Theta(\sqrt d)\), and resolves the Euclidean Steinitz–Bergström conjecture.

Every infinite-dimensional real Banach space contains a compact doubling subset that admits no bi-Lipschitz embedding into any finite-dimensional real normed space. The doubling constant has a universal bound, independent of the ambient Banach space. This answers the Lang–Plaut problem negatively, even for compact subsets of Hilbert space.

We give two independent constructions of binary words of one length at most d whose ordinary edit-distance metrics require ℓ₁ distortion \(\exp(\Omega(\sqrt{\log d\,\log\log d}))\) for every sufficiently large d. We also prove a constant-distortion binary conversion for one prescribed input length. Together with the companion upper embedding theorem, these lower bounds determine the order of logarithmic distortion uniformly over finite alphabets with at least two symbols.

We give two finite-circle constructions of binary strings whose least ℓ₁ distortion is \(\exp(\Omega(\sqrt{\log d\,\log\log d}))\), where d bounds their length. Both constructions supply words of one common length for every sufficiently large cap. Two direct binary coding arguments transfer the constructions with absolute distortion and logarithmic block width. We also develop the overlapping-substring method of Ostrovsky and Rabani into a complete finite histogram embedding at the same exponential scale, uniformly over all finite alphabets and all words of length at most d, including the empty word.

We determine the exponential scale of the least ℓ₁ distortion of unit-cost edit distance on all strings of length at most d. For every sufficiently large d, uniformly over finite alphabets of size at least two, the distortion lies between \(\exp(c\sqrt{\log d\,\log\log d})\) and \(\exp(C\sqrt{\log d\,\log\log d})\) for absolute constants \(c,C\gt 0\). The lower bound already holds on binary strings of one common length. Thus the order of logarithmic distortion is sharp up to absolute constants.

The two-cylinder covering of a regular tetrahedron can be perturbed to give finite covers with total perpendicular base area strictly below half its minimum projection area. These covers give negative answers to both the half-area question and the directionwise normalized half-bound conjecture. By affine invariance, the directionwise conclusion holds for every nondegenerate tetrahedron.

We prove that radially aligned segment sweeps admit finite cylinder covers with one triangular base for each interval of any tagged partition. As the mesh tends to zero, the total perpendicular base area converges to a weighted parameter area. An explicit application covers every regular tetrahedron with total base area below half its minimum projection area, giving negative answers to the half-area question and the directionwise normalized 1-Codimensional Cylinder Covering Conjecture.

We approximate compact ruled families of segments by finitely many cylinders with square intercept tiles and perpendicular-base area at most their integral projection cost plus any positive error. The velocity field is C¹, and its differential has opposite real eigenvalues whose magnitudes are strictly below the inverse segment half-length; both eigenvalues may vanish. The result includes square-zero differentials with unrestricted shear and fields that pass between the two regimes. It holds for arbitrary compact label sets.

A regular tetrahedron admits a finite cylinder covering with compact triangular perpendicular bases whose total area is less than half its minimum orthogonal projection area. This disproves the half-area cylinder-covering conjecture. By affine invariance, the same construction gives a counterexample to the directionwise normalized half-bound for every nondegenerate tetrahedron.

We prove the strong isotropic constant conjecture: in each dimension, simplices are the unique maximizers of the isotropic constant among convex bodies. We also prove the sharp entropy bound \(h(f)\ge m+\tfrac12\log\det\mathop{\mathrm{Cov}}\nolimits (f)\) for every log-concave probability density on ℝ^m, with equality precisely for invertible affine images of products of one-sided exponential laws.