Towards sparse optimization over convex loops: Equivalence of Square Root Velocity distance and Wasserstein-Fisher-Rao

The Wasserstein-Fisher-Rao (WFR) distance on $S^{2}$ has recently been shown to coincide with a classical elastic distance between $S^{2}$-immersions in the theory of Riemannian shape analysis. While this correspondence holds in dimension $2$, the analogous statement fails in general on $S^{1}$ and, in the case of convex curves, it cannot be derived from existing two-dimensional arguments. In this paper, we establish that for convex absolutely continuous immersions of $S^{1}$ in the plane, the shape distance induced by the square root velocity transformation (SRVT) is indeed equivalent to the WFR distance acting on their associated length measures. The proof exploits a monotonicity principle for optimal transport on the universal cover of the circle, which in turn guarantees the existence of an optimal reparametrization achieving the SRVT infimum and enables a one-dimensional unbalanced optimal transport reformulation. Motivated by this equivalence, we further investigate the role of sparsity in shape optimization problems formulated in terms of length measures and regularized by the WFR distance. We study linear optimization over the corresponding balls, for which we prove a finiteness result when the reference measure is discrete, and propose a convex, positively one-homogeneous regularizer suitable for conditional gradient algorithms.