Injectivity radius and geometric bound on Kendall shape space

Abstract

We compute the injectivity radius on particular subspaces of the Kendall shape space $\Sigma_{m}^{k}$. The space $\Sigma_{m}^{k}$ is useful for representing the shapes associated to collections of $k$ vector columns in $\mathbb{R}^{m}$. We determine also an upper bound of arc length parameter of specific curves lying in particular subspaces of the pre-shape sphere $\mathcal{S}_{m}^{k}$. Here, $\Sigma_{m}^{k}$ is the quotient space of $\mathcal{S}_{m}^{k}$ modulo the special
orthogonal group $SO\left(m\right)$ acting on the left.