Isoperimetric Density and Ulam's Packing Conjecture: A Curvature-Variance Proof via the Gauss-Bonnet Theorem

Ulam's packing conjecture asserts that the ball is the convex body in R^3 with the lowest maximum packing density. We present a proof grounded in two classical results: the isoperimetric inequality and the Gauss-Bonnet theorem. The central observation is that packing density is controlled by the minimum centre-to-centre distance achievable between non-overlapping copies of a body: when this distance falls below 2R_eq (the diameter of the equal-volume sphere), packing density exceeds that of the sphere. We show that for any non-spherical convex body, positive curvature variance implies the existence of an interlocking configuration in which the minimum centre-to-centre distance is strictly less than 2R_eq, and that the sphere uniquely achieves the maximum minimum distance of exactly 2R. The argument is self-contained, requiring only the isoperimetric inequality, the Gauss-Bonnet theorem, the Alexandrov uniqueness theorem, and a Taylor expansion of boundary curvature.