I spent a year of my PhD stuck on a 2002 problem of Schechtman. GPT 5.5-Pro helped me finish: vector balancing for zonotopes (shadows of a cube)!
For any zonotope Z ⊂ ℝᵈ, v₁,...,vₙ ∈ Z, there are signs x₁,...,xₙ ∈ {-1, 1} with x₁v₁ ... xₙvₙ ∈ O(√d) Z, sharp. [1/4]