This paper consists of two halves. In the first half of the paper, we consider real‐valued functions f whose domain is the vertex set of a graph G and that are Lipschitz with respect to the graph distance. By placing a uniform distribution on the vertex set, we treat as a random variable. We investigate the link between the isoperimetric function of G and the functions f that have maximum variance or meet the bound established by the subgaussian inequality. We present several results describing the extremal functions, and use those results to (a) resolve a conjecture by Bobkov, Houdré, and Tetali characterizing the extremal functions of the subgaussian inequality of the odd cycle and (b) provide a construction that gives a partial negative answer to a question by Alon, Boppana, and Spencer on the relationship between maximum variance functions and the isoperimetric function of product graphs. While establishing a discrete analogue of the curved Brunn‐Minkowski inequality for the discrete hypercube, Ollivier and Villani suggested several avenues for research. We resolve them in the second half of the paper as follows:

• They propose that a bound on t ‐midpoints can be obtained by repeated application of the bound on midpoints, if the original sets are convex. We construct a specific example where this reasoning fails, and then prove our construction is general by characterizing the convex sets in the discrete hypercube.
• A second proposed technique to bound t ‐midpoints involves new results in concentration of measure. We follow through on this proposal, with heavy use on results from the first half of the paper.
• We show that the curvature of the discrete hypercube is not positive or zero, but we also give a result indicating that it may satisfy a weaker version of curvature.

中文翻译：

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正曲线图

本文分为两半。在本文的上半部分，我们考虑实值函数f，它们的域是图G的顶点集，相对于图距离为Lipschitz。通过在顶点集上放置均匀分布，我们将其视为随机变量。我们研究了G的等距函数与函数f之间的联系具有最大方差或满足亚高斯不等式所确定的界限。我们提供了一些描述极值函数的结果，并将这些结果用于（a）解决Bobkov，Houdré和Tetali提出的奇数周期次高斯不等式极值函数的猜想，以及（b）提供部分给出的构造Alon，Boppana和Spencer对产品图的最大方差函数与等参函数之间的关系的否定回答。在为离散超立方体建立弯曲的Brunn-Minkowski不等式的离散模拟时，Ollivier和Villani建议了几种研究途径。我们在本文的后半部分中解决这些问题，如下所示：

• 他们建议，如果原始集合是凸的，则可以通过重复应用中点上的界线来获得t中点上的界线。我们构造了一个推理失败的具体示例，然后通过表征离散超立方体中的凸集来证明我们的构造是通用的。
• 提出的第二种限制t中点的技术涉及到测量集中的新结果。我们遵循了该建议，并大量使用了本文上半部分的结果。
• 我们表明离散超立方体的曲率不是正值或零，但是我们也给出了表明它可能满足较弱曲率的结果。