AI 中文总结
研究反向切格不等式，将其从凸集扩展，通过主宽度概念分析凸体最大化序列，证得三维中特定主宽度比消失的凸体序列是最大化的，任意维度类菱形集也有此性质。
AI 中文摘要
我们研究反向切格不等式，它从上方界定了第一狄利克雷拉普拉斯特征值与切格常数平方的比值。首先将此不等式从凸集扩展到更广泛的类。接着分析凸体中的最大化序列，为量化区域坍缩引入主宽度概念。对于三维凸体，证明若第一主宽度与第二主宽度之比沿凸体序列消失，则该序列是最大化的。最后在任意维度证明对于类菱形集同样性质成立。
英文摘要
We study the reverse Cheeger inequality, which bounds from above the ratio of the first Dirichlet Laplacian eigenvalue to the square of the Cheeger constant. We first extend this inequality from convex sets to a broader class. We then analyze maximizing sequences among convex bodies. To quantify domain collapse, we introduce the notion of principal widths. For three-dimensional convex bodies, we prove that if the ratio of the first principal width to the second principal width vanishes along a sequence of convex bodies, then such sequence is maximizing. Finally, in arbitrary dimensions, we prove that the same property holds for the class of rhomboid-like sets.
Comments32 pages
