体积与投影不等式 II：行列式与$L_p$-和
Volume and Projection Inequalities II: Determinants and $L_p$-Sums
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中文总结 AI 辅助
受Dembo--Cover--Thomas猜想启发，研究正交投影体积与Firey $L_p$-和及行列式幂的不等式，证明$1<p<2$时强双项投影不等式在$n\geq2$维均不成立，同时分类了不同维度与p值范围下各类不等式的成立情况。
中文摘要 AI 辅助
受Dembo--Cover--Thomas猜想的启发，我们研究了正交投影体积的不等式及其与Firey $L_p$-和的关系，以及它们的行列式幂类似形式。对于$L_p$-zonoid（$L_p$类zonoid体）$K,L\subset\mathbb{R}^n$和$u\in S^{n-1}$，我们考虑如下不等式：\\[ \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 . \\] 对于任意$1<p<2$，我们证明该不等式在所有维度$n\geq2$下均不成立。相比之下，省略右侧第二项得到的弱单项不等式，在二维空间中对$1\leq p\leq2$的整个范围均成立。该平面结果的证明用到了归一化对偶映射的一个精确估计。我们还对$0<p<2$范围内对应的行列式幂不等式进行了分类：强双项不等式在二维成立，在所有$n\geq3$的维度不成立；弱单项不等式在$n\leq3$维下对$0<p\leq1$成立，在$n\geq4$时不成立；对于$1<p<2$，它仅在二维成立。
英文摘要
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 $n\geq2$. 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 $1\leq p\leq2$. 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 $n\geq3$. The weak one-term inequality holds for $0<p\leq1$ in dimensions $n\leq3$ and fails for $n\geq4$; for $1<p<2$, it holds only in dimension two.
