In learning methods defined on a manifold, it is often important to embed the manifold into a Hilbert space to apply kernel methods, as it enables the utilization of linear techniques in the transformed space, thereby facilitating complex data analysis and classification tasks with nonlinear structures [10]. In [6], the authors define an embedding of the \(\textit{m}\) -dimensional Kendall shape space into an \(\mathbb {R}^{\textit{N}}\) space, from which kernel methods are introduced in the Kendall shape space. In this paper, we significantly simplify this result by achieving an embedding of the Kendall shape space into a Euclidean space \(\mathbb {R}^{\textit{n}}\) of much lower dimension \(\textit{n}\) than \(\textit{N}\) , greatly simplifying calculations and computational costs of applications. Additionally, we characterize the image of the Kendall shape space in \(\mathbb {R}^{\textit{n}}\) , allowing us to define a new extrinsic mean of shapes in the Kendall shape space.
