I will present a historical account of some work of Schoenberg
in metric geometry: from his metric space embeddings into
Euclidean space and into spheres (*Ann. of Math.* 1935), to his
characterization of positive definite functions on spheres
(*Duke Math. J.* 1942). It turns out these results can be
viewed alternately in terms of matrix positivity: from appearances
of (conditionally) positive matrices in analysis, to the
classification of entrywise positivity preservers in all
dimensions.

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Last updated: 09 Apr 2019