Quermassintegral maximizers in John’s position (simplex/cube conjecture)

Establish that for every n ≥ 1 and every convex body K in E^n whose largest-volume inscribed ellipsoid is the Euclidean unit ball (John’s position), the quermassintegrals satisfy W_j(K) ≤ W_j(Δ_n) for all 0 ≤ j ≤ n, where Δ_n is the dilation of the regular simplex T_n with unit inscribed ball. In the centrally symmetric case, prove that for every K in John’s position, W_j(K) ≤ W_j(C_n) for all 0 ≤ j ≤ n, where C_n = [-1,1]^n is the cube circumscribed about the unit ball.

Background

The paper leverages bounds on |K + B^n| via Steiner’s formula and quermassintegrals W_j(K) when K is in John’s position. Known extremal results show that, in John’s position, volume (W_0) and mean width (W_{n-1}) are maximized by the regular simplex in the general case and by the cube in the centrally symmetric case.

Remark 3 formulates a broader conjecture asserting that these extremizers should also maximize every quermassintegral W_j across all indices j, extending the currently verified cases j ∈ {0,1,n−1} (and the trivial equality for j = n).