A quadratic comparison of Neumann eigenvalues on thin convex domains in arbitrary dimenstion

Let $\mathrm{\Omega }\subset {\mathbb{R}}^n$ be a bounded convex domain that is thin around a chosen diameter segment. We compare its Neumann spectrum with the spectrum of that segment weighted by the $(n-1)$-dimensional volumes of its perpendicular sections. We prove an $O({\epsilon }^2)$ comparison of the mean-zero inverse operators and, consequently, an $O({\epsilon }^2)$ eigenvalue comparison for every fixed index in every dimension $n\ge 2$. The constants depend only on the dimension and the eigenvalue index. Thin rectangles show that the quadratic exponent is optimal.