In this paper, we derive variational formulas for the asymptotic exponents (i.e., convergence rates) of the concentration and isoperimetric functions in the product Polish probability space under certain mild assumptions. These formulas are expressed in terms of relative entropies (which are from information theory) and optimal transport cost functionals (which are from optimal transport theory). Hence, our results verify an intimate connection among information theory, optimal transport, and concentration of measure or isoperimetric inequalities. In the concentration regime, the corresponding variational formula is in fact a dimension-free bound in the sense that this bound is valid for any dimension. A cardinality bound for the alphabet of the auxiliary random variable in the expression of the asymptotic isoperimetric exponent is provided, which makes the expression computable by a finite-dimensional program for the finite alphabet case. We lastly apply our results to obtain an isoperimetric inequality in the classic isoperimetric setting, which is asymptotically sharp under certain conditions. The proofs in this paper are based on information-theoretic and optimal transport techniques.

翻译：本文在一定的温和假设下，推导了乘积波兰概率空间中集中函数与等周函数的渐近指数（即收敛速率）的变分公式。这些公式通过相对熵（源自信息论）和最优传输成本泛函（源自最优传输理论）来表达。因此，我们的结果验证了信息论、最优传输以及测度集中或等周不等式之间的紧密联系。在集中性体系中，相应的变分公式实际上是一个与维度无关的界，因为该界对任意维度均成立。我们还给出了渐近等周指数表达式中辅助随机变量字母表基数的一个界，这使得在有限字母表情形下，该表达式可通过有限维规划进行计算。最后，我们应用所得结果推导了经典等周设定下的一个等周不等式，该不等式在特定条件下是渐近尖锐的。本文的证明基于信息论与最优传输技术。