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.
We prove a sharp isoperimetric inequality for the harmonic mean of the first $n$ nonzero Neumann eigenvalues of the Witten-Laplacian on origin-symmetric Lipschitz domains in space forms, endowed with radial log-concave measures. The main novelty is that we establish the sharp harmonic mean inequality under general radial log-concave measures, without assuming the weight function to be non-increasing. This extends previous results that were restricted to specific or more restrictive weighted settings. The proof relies on a refined analysis of the first eigenfunction on geodesic balls, a monotonicity property derived from a convexity condition on the radial weight, and a matrix trace inequality.
We establish a sharp lower bound for the total Gauss-Kronecker curvature of convex hypersurfaces in Cartan-Hadamard manifolds with pinched negative curvature. The bound holds in all dimensions when the diameter is small relative to the curvature scale, and in dimensions 4 and 5 without any restriction on the diameter, provided that the pinching is sufficiently tight. The proofs are based on the Chern-Gauss-Bonnet theorem and weighted Hsiung-Minkowski inequalities. As an application, we obtain the isoperimetric inequality of the Cartan-Hadamard conjecture in dimension 5 under sufficiently pinched curvature.
We say that a convex body is in Faber-Krahn position if it minimizes the first Dirichlet eigenvalue within its volume-preserving linear orbit. We prove that this position is unique up to orthogonal transformations, answering a question of Schmuckenschlaeger from 2011. This is a corollary of a new log-convexity property of the first eigenvalue under positive definite linear deformations. While the centrally symmetric case follows from the Gaussian B-theorem, the extension to arbitrary convex bodies requires a quantitative analysis of conditioned Brownian motion. As consequences, we obtain a new proof of the Polya-Szego theorem for triangles and its analogue for simplices, and show that regular polygons minimize the first eigenvalue within their linear orbits of fixed volume. We also prove related convexity results for the first eigenvalue of the Ornstein-Uhlenbeck operator, the inverse inradius and the planar Cheeger constant. In a different direction, we show using similar ideas that the Gaussian conjugate Rogers-Shephard inequality due to Milman-Nakamura-Tsuji yields improved Schmuckenschlaeger-type bounds for intersections and Minkowski sums of centrally symmetric convex bodies.
We prove a sharp comparison, with Obata-type rigidity, for all Neumann eigenvalues of one-dimensional $\mathrm{CD}(1,2)$ spaces against the Legendre model, under a convexity condition on the density. It follows that minimizers of the $k$-th Laplace eigenvalue among closed surfaces of Gaussian curvature at least $1$ cannot collapse in the measured Gromov-Hausdorff completion, for every $k\ge2$. We also give a variational proof that smooth minimizers are round.
We prove the Weinstock inequality for the first Steklov eigenvalue of convex domains in hyperbolic space $\mathbb{H}^{n}$, resolving Open Question 4.27 of Colbois-Girouard-Gordon-Sher(2024) for the remaining case $n=3$. Our argument replaces the global monotonicity required in earlier work Gu-Li-Wan(2025) with a one-crossing property, which is established via an explicit slope comparison. The proof works uniformly for all $n\geq 3$.
We study extremizers for a trilinear Stein-Weiss inequality on $\mathbb{R}^n$. Within the known boundedness region, we prove attainment under two additional assumptions: all six weight exponents are nonnegative, and at least one pair of Lebesgue exponents is admissible. The proof combines symmetric decreasing rearrangement with a logarithmic radial reduction to a translation-invariant bilinear operator on $\mathbb{R}$ whose kernel belongs to $L^1\left(\mathbb{R}^2\right)$. A common-scale compactness argument rules out relative separation of the two arguments and yields norm attainment. We then derive the Euler-Lagrange system. In the fully symmetric case, every normalized nonnegative extremizing triple is diagonal. Finally, we establish the origin-centered Kelvin invariance of the resulting scalar equation at the scaling exponent and record the unweighted conformal example.
