Convex sets with vanishing relative width and the reverse Cheeger inequality
arXiv:2607.25708
The paper extends the reverse Cheeger inequality beyond convex sets, introduces principal widths to measure domain collapse, and shows that sequences of convex bodies with vanishing ratio of first to second principal width maximize the inequality, with analogous results for rhomboid-like sets in any dimension.
Abstract
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.
32 pages