We prove various extensions of the Loomis–Whitney inequality and its dual, where the subspaces on which the projections (or sections) are considered are either spanned by vectors $w_i$ of a not necessarily orthonormal basis of ${\mathbb{R}}^n$, or their orthogonal complements. In order to prove such inequalities, we estimate the constant in the Brascamp–Lieb inequality in terms of the vectors $w_i$. Restricted and functional versions of the inequality will also be considered.

中文翻译：

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关于仿射不变量和局部 Loomis-Whitney 型不等式

我们证明了 Loomis-Whitney 不等式及其对偶的各种扩展，其中考虑投影（或部分）的子空间要么由向量跨越 ${瓦}_{\mathrm{一世}}$ 不一定是正交基 ${\mathbb{电阻}}^n$，或它们的正交补。为了证明这种不等式，我们根据向量估计 Brascamp-Lieb 不等式中的常数${瓦}_{\mathrm{一世}}$. 不等式的受限版本和函数版本也将被考虑。