The conjectured monotonicity I_{m,-n}(K)^{1/m} ≥ I_{k,-n}(K)^{1/k} fails for 1 ≤ m < k ≤ n-1 when n > (m+2)(k+2)-2, but the chain I_{1,-3} ≥ I_{2,-3}^{1/2} ≥ I_{3,-3}^{1/3} = 1 holds in dimension three.
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On the monotonicity of affine quermassintegrals
The conjectured monotonicity I_{m,-n}(K)^{1/m} ≥ I_{k,-n}(K)^{1/k} fails for 1 ≤ m < k ≤ n-1 when n > (m+2)(k+2)-2, but the chain I_{1,-3} ≥ I_{2,-3}^{1/2} ≥ I_{3,-3}^{1/3} = 1 holds in dimension three.