# Convex and metric geometry > 15 AI-made results in convex and metric geometry, each explained in plain English with its trust label, Lean status and sources. Canonical: https://closedform.nishachay.tech/science/mathematics/convex-and-metric-geometry/ Last updated: 2026-10-14 15 results in OpenAI Math 2026. 15 AI-made results in convex and metric geometry, from 30 papers in the OpenAI math collection, each explained in plain English. 13 have a formal Lean proof listed; 2 are claimed in a paper and not yet machine-checked. The oldest question here dates back to 1939: [The Mahler conjectures, functional inequalities and polar-product symplectic width](https://closedform.nishachay.tech/e/openai-math-2026-087/). ## Results (15) - [The Mahler conjectures, functional inequalities and polar-product symplectic width](https://closedform.nishachay.tech/e/openai-math-2026-087/) - Entry 087 · Symmetric convex bodies cannot beat Mahler's volume bound. (Formal proof listed) - [Sharp projection-body inequalities and a counterexample to simplex maximization](https://closedform.nishachay.tech/e/openai-math-2026-088/) - Entry 088 · In four or more dimensions, ellipsoids have the smallest shadow body for their volume. (Formal proof listed) - [Bounded-distortion L 1 embeddings of planar and bounded-treewidth graphs](https://closedform.nishachay.tech/e/openai-math-2026-089/) - Entry 089 · Every planar road network can be redrawn in taxicab distance with stretch below one fixed number. (Formal proof listed) - [Triangular-lattice optimality, long-range Riesz and Coulomb energies, and spherical logarithmic energy](https://closedform.nishachay.tech/e/openai-math-2026-090/) - Entry 090 · No arrangement of points in the plane beats the triangular lattice for any steadily fading repulsion. (Formal proof listed) - [Logarithmic and L p Brunn–Minkowski inequalities and the B-conjecture](https://closedform.nishachay.tech/e/openai-math-2026-091/) - Entry 091 · Blending two symmetric shapes the thrifty way never loses volume below their geometric mean. (Formal proof listed) - [The optimal order of convex-body covering density](https://closedform.nishachay.tech/e/openai-math-2026-092/) - Entry 092 · The worst convex shape needs covering density about n log n in dimension n, lattice or not. (Formal proof listed) - [Dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures](https://closedform.nishachay.tech/e/openai-math-2026-093/) - Entry 093 · Log-concave laws with bell-curve tails in every direction obey a dimension-free log-Sobolev bound, the paper claims. (Claimed, not checked) - [Subpolynomial dimension reduction in L p](https://closedform.nishachay.tech/e/openai-math-2026-094/) - Entry 094 · Points in Lp space can be squeezed into far fewer coordinates if distances may stretch slightly. (Formal proof listed) - [Hyperbolicity cones without semidefinite lifts](https://closedform.nishachay.tech/e/openai-math-2026-095/) - Entry 095 · A claimed proof that some hyperbolicity cones have no semidefinite description, even with hidden variables. (Formal proof listed) - [The Gaussian propeller conjecture in every dimension](https://closedform.nishachay.tech/e/openai-math-2026-096/) - Entry 096 · Three 120-degree wedges are the best way to cut a bell-curve cloud, in every dimension. (Formal proof listed) - [The Euclidean Steinitz–Bergström bound](https://closedform.nishachay.tech/e/openai-math-2026-097/) - Entry 097 · One choice of signs keeps every running total of unit vectors within a constant times √d, at any length. (Formal proof listed) - [Compact counterexamples to bi-Lipschitz dimension reduction](https://closedform.nishachay.tech/e/openai-math-2026-098/) - Entry 098 · Some uncrowded point sets in infinite dimensions cannot be copied faithfully into any finite dimension. (Formal proof listed) - [The sharp exponential scale of edit-distance distortion](https://closedform.nishachay.tech/e/openai-math-2026-099/) - Entry 099 · Edit distance between words can be drawn as ℓ1 distance only with distortion of size exp(√(log d·log log d)). (Formal proof listed) - [Cylinder coverings below the half-area bound](https://closedform.nishachay.tech/e/openai-math-2026-100/) - Entry 100 · Cylinders can cover a regular tetrahedron for less than half its smallest shadow. (Formal proof listed) - [The sharp simplex conjecture for isotropic constants](https://closedform.nishachay.tech/e/openai-math-2026-101/) - Entry 101 · In every dimension, simplices alone have the largest isotropic constant among convex bodies, the paper claims. 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