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The title of this post is also the title of my most recent paper on the arXiv, from 2019, with Joachim Hilgert of the University of Paderborn. The published version is titled Spectral Properties of Convex Bodies , Journal of Lie Theory 30 (2020) 315-355.
Here's a preprint version of the J Lie Theory article:
Compared to the arXiv version, the J Lie Theory version (and the above-linked preprint) has less detail about the background results by other authors (mainly Jiri Dadok's theory of polar representations of compact Lie groups, and the Madden-Robertson theory of regular convex bodies) which we use, and a bit more detail in the proof of our main result. It also has a more extensive discussion of infinite-dimensional Jordan algebraic systems, in the context of discussion of ways in which the Jordan algebraic sy
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