Volume and Projection Inequalities II: Determinants and $L_p$-Sums

Abstract

We study inequalities for the volume of orthogonal projections and their relation to Firey $L_p$-sum, together with their determinant-power analogues, motivated by the Dembo--Cover--Thomas conjecture. For $L_p$-zonoids $K,L\subset {\mathbb{R}}^n$ and $u\in S^{n-1}$, we consider the inequality \[ \left( \frac{|K\oplus_p L|} {|P_{u^\perp}(K\oplus_p L)|} \right)^p \geq \left( \frac{|K|}{|P_{u^\perp}K|} \right)^p + \left( \frac{|L|}{|P_{u^\perp}L|} \right)^p . \] For every $1<p<2$, we prove that this inequality fails in every dimension $n\ge 2$. In contrast, the weak one-term inequality, obtained by omitting the second term on the right-hand side, holds in dimension two throughout the full range $1\le p\le 2$. The proof of this planar result uses a sharp estimate for the normalized duality map. We also classify the corresponding determinant-power inequalities in the range $0<p<2$. The strong two-term inequality holds in dimension two and fails in every dimension $n\ge 3$. The weak one-term inequality holds for $0<p\le 1$ in dimensions $n\le 3$ and fails for $n\ge 4$; for $1<p<2$, it holds only in dimension two.