Prefix admissibility, bounded calibrations, and optimal size-based menus for uniform values

A seller offers N goods to one buyer with additive values that are independent and uniform on [0,1]. Giannakopoulos and Koutsoupias proposed a deterministic menu for this problem, the Straight-Jacket Auction, which prices bundles by size only. They proved it revenue-optimal for up to six goods and conjectured optimality for every N. We prove a geometric criterion that yields optimality in higher dimensions. The allocation regions of such menus are products of SIM bodies, a family of polytopes that are generalized permutahedra. We show that a SIM body whose prefix bodies all have nonnegative deficiency minimizes the anisotropic perimeter with density ‖ν‖₁ minus twice the volume among its subsets, and that it is calibrable when it is balanced. The proof verifies the boundary-curvature condition of Caselles, Chambolle, Moll and Novaga by induction over the facets, which are again products of permutahedra and SIM bodies. Combined with a calibration-based revenue dual, the criterion shows that a size-based menu is optimal among all incentive-compatible, individually rational mechanisms, including randomized ones, as soon as finitely many polynomial sign conditions hold. We verify these conditions with exact rational arithmetic for every N ≤ 50. The optimal menus omit several large bundle sizes. We prove that the number of omitted sizes is O(N3/4 (log N)1/4) for every menu of this form, so it is sublinear in N. Whether such menus exist for every N remains open.