Exponential bounds for the spherical Blaschke-Lebesgue problem

Abstract

The Blaschke-Lebesgue theorem states that the Reuleaux triangle has the smallest area among planar convex bodies of a fixed constant width. We study how small bodies of constant width can be on the unit sphere $\mathbb S^n$ when $n$ is large. For a spherical convex body $K\subset \mathbb S^n$ of constant width $w\in(0,\pi)$, its relative effective radius is \[ \left(\frac{\mu_n(K)}{\mu_n(\mathbb B^n(w/2))}\right)^{1/n}, \] where $\mu_n$ is the spherical $n$-measure and $\mathbb B^n(w/2)$ is a geodesic ball of radius $w/2$. Let $\sigma_n(w)$ be the infimum of the relative effective radius over all spherical bodies of constant width $w$. Define $\underline{\sigma}(w)=\liminf_{n\to\infty}\sigma_n(w)$ and $\overline{\sigma}(w)=\limsup_{n\to\infty}\sigma_n(w)$. For each fixed $w\in(0,\pi)$, we prove non-trivial bounds \[ 0<\sigma_{\ell}(w)\le \underline{\sigma}(w)\le \overline{\sigma}(w)\le \sigma_u(w)<1, \] where $\sigma_\ell(w)$ and $\sigma_u(w)$ are defined in terms of $w$ either explicitly or through a root of a quartic equation. The upper bounds are obtained by constructing small spherical bodies of constant width: for $w\le\pi/2$ by a spherical version of the recent Arman-Bondarenko-Nazarov-Prymak-Radchenko Euclidean example, and for $w>\pi/2$ by spherical duality. The lower bounds combine a spherical adaptation of Schramm's illumination argument with a Gaussian autocorrelation method for the associated convex cones.