On the monotonicity of affine quermassintegrals

Lutwak's affine quermassintegral theory is a foundational component of modern affine Brunn--Minkowski theory. Developed in the 1980s, it provides affine analogues of the classical quermassintegrals and has led to a rich family of sharp affine isoperimetric inequalities. A central question in this program, going back to Lutwak's 1988 work, is an Alexandrov--Fenchel-type monotonicity principle for the normalized $L^{-n}$-moment quermassintegrals $I_{k,-n}$. In one form, this principle predicts that $$I_{m,-n}(K)^{1/m}\ge I_{k,-n}(K)^{1/k}, \qquad 1\le m<k\le n .$$ The question was recorded in Gardner's 2006 book Geometric Tomography as part of its problem list, and the comparison with the top dimension, $k=n$, was established by Milman and Yehudayoff in their 2023 JAMS paper. We show that the proposed monotonicity does not persist in the full range. More precisely, for every triple of integers $m,k,n$ satisfying $1\le m<k\le n-1$ and $n>(m+2)(k+2)-2$, there exists an origin-symmetric $C^2_+$ convex body $K\subset\mathbb R^n$ such that $$I_{m,-n}(K)^{1/m} < I_{k,-n}(K)^{1/k}.$$ The example is obtained from the Euclidean ball by an arbitrarily small degree-four spherical harmonic perturbation. On the positive side, we prove that the endpoint chain is true in dimension three: for every convex body $K\subset\mathbb R^3$, $$I_{1,-3}(K)\ge I_{2,-3}(K)^{1/2}\ge I_{3,-3}(K)^{1/3}=1.$$ The equality cases in both non-trivial inequalities are exactly ellipsoids, up to translation and nonsingular affine transformations.