If $f, g, h\colon \mathbb{R}^n\longrightarrow\mathbb{R}_{\geq 0}$ are non-negative measurable functions such that $h(x+y)$ is greater than or equal to the $p$-sum of $f(x)$ and $g(y)$, where $-1/n\leq p\leq\infty$, $p\neq0$, then the Borell–Brascamp–Lieb inequality asserts that the integral of $h$ is not smaller than the $q$-sum of the integrals of $f$ and $g$, for $q=p/(np+1)$.

In this paper we obtain a discrete analog for the sum over finite subsets of the integer lattice $\mathbb Z^n$: under the same assumption as before, for $A,B\subset\mathbb{Z}^n$}, then $\sum_{A+B}h\geq[(\sum_{\mathrm {r}{f}(A)} f)^q+(\sum_B g)^q]^{1/q}$, where $\mathrm{r}_{\!f}(A)$ is obtained by removing points from $A$ in a particular way, and depending on $f$. We also prove that the classical Borell–Brascamp–Lieb inequality for Riemann integrable functions can be obtained as a consequence of this new discrete version.

中文翻译：

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关于离散的Borell–Brascamp–Lieb不等式

如果$ f，g，h \冒号\ mathbb {R} ^ n \ longrightarrow \ mathbb {R} _ {\ geq 0} $是非负可测量函数，则$ h（x + y）$大于或等于$ f（x）$和$ g（y）$的$ p $ -sum，其中$ -1 / n \ leq p \ leq \ infty $，$ p \ neq0 $，然后是Borell–Brascamp– Lieb不等式断言，对于$ q = p /（np + 1）$，$ h $的积分不小于$ f $和$ g $的积分的$ q $-和。

在本文中，我们为整数格$ \ mathbb Z ^ n $的有限子集上的总和获得了一个离散模拟，在与以前相同的假设下，对于$ A，B \ subset \ mathbb {Z} ^ n $}，然后$ \ sum_ {A + B} h \ geq [（\ sum _ {\ mathrm {r} {f}（A）} f）^ q +（\ sum_B g）^ q] ^ {1 / q} $，其中$ \ mathrm {r} _ {\！f}（A）$是通过以特定方式并根据$ f $从$ A $中删除点而获得的。我们还证明，由于这种新的离散形式，可以得到Riemann可积函数的经典Borell-Brascamp-Lieb不等式。