Optimal Lipschitz constant for planar Minkowski stability

Determine the optimal universal constant in the Steiner-centered planar Minkowski stability estimate d_H(K_\mu,K_\nu)\le C W_1(\mu,\nu) for balanced, nondegenerate probability measures on the circle, under the normalization S_{K_\mu}=2\pi\mu and S_{K_\nu}=2\pi\nu.

Background

The paper establishes a quantitative inverse-stability estimate for planar Minkowski reconstruction: if balanced, nondegenerate probability measures \mu and \nu on the circle are reconstructed as Steiner-centered convex bodies K_\mu and K_\nu with surface-area measures 2\pi\mu and 2\pi\nu, then d_H(K_\mu,K_\nu)\le\pi W_1(\mu,\nu). The constant \pi follows from an explicit Green kernel for the operator D^2+1 and the fact that the kernel has geodesic Lipschitz norm 1/2.

The paper also proves that the linear dependence on W_1 is sharp in exponent, using smooth positive perturbations of the uniform measure. However, it does not determine whether the constant \pi is optimal, nor what the smallest admissible universal constant is.