Affine quermassintegral conjecture over real, complex, and quaternionic spaces

Establish that for every convex body K in F^n with non-empty interior, where F is one of the real numbers R, the complex numbers C, or the quaternions H, and for every integer m with 1 ≤ m ≤ n, the inequality |K|^{-m} ≥ ((κ_{mp})^n/(κ_{np})^m) ∫_{Gr_m(n,F)} |P_E K|^{-n} dE holds, with equality if and only if K is a real, complex, or quaternionic ellipsoid. Here p = dim_R F, Gr_m(n,F) is the Grassmannian of m-dimensional F-linear subspaces of F^n, P_E denotes orthogonal projection onto E, and κ_d is the volume of the d-dimensional Euclidean unit ball.

Background

The paper extends Busemann’s random simplex and intersection inequalities to complex and quaternionic vector spaces and develops associated invariance properties under SL(n,F). Within this framework, the authors formulate an analogue of Lutwak’s conjecture (recently confirmed by Milman–Yehudayoff in the real case) for all three fields F ∈ {R, C, H}.

The conjecture asserts a sharp inequality comparing the volume of K with an integral involving projections onto m-dimensional subspaces. It characterizes ellipsoids as the equality cases. The authors note that while a symmetrization-based approach works in the real case, it fails in the complex and quaternionic settings, indicating that new methods are required. They also verify the conjecture in a special case (m=1 with K the unit ball of a complex or quaternionic norm).