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This paper considers the general problem of the analysis of groups of shapes, and the issue of correspondence in that context. Many papers have been published on the topic of pairwise shape distances and pairwise shape similarity measures. However, most of these approaches make the implicit assumption that the methods developed for pairs of shapes are then sufficient when it comes to the problem of analyzing groups of shapes. In this paper, we consider the general case of pairwise and groupwise shape analysis within an infinite-dimensional Riemannian framework. We show how the issue of groupwise or pairwise shape correspondence is inextricably linked to the issue of the metric. We discuss how data-driven approaches can be used to find the optimum correspondence, and demonstrate how different choices of objective function lead to different groupwise correspondence, and why this matters in terms of groupwise modelling of shape.