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Let $Q \in \operatorname{Conv} SO(3)$.

Is there a way to retrieve an explicit representation of $Q$ as convex combination $Q=\sum_{k=1}^{r}{\lambda_{i}R_{i}}, R_{i} \in SO(3)$?

An approximation can also be useful.

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A closely related answer: The convex hull of SO(n) is studied in this paper by Saunderson, Parrilo, and Willsky (2015). In particular, that paper proves that this convex hull is doubly spectrahedral, and provides explicit representations. For $n=3$, see (1.11) in the paper.

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  • $\begingroup$ That's where I'm coming from :):):). I'm relaxing a SO(3) optimization problem using their representation and get something in Conv SO(3). But I still need to obtain an answer in SO(3) so I thought I'd try to decompose the "something" into SO(3) "bits" and use one of the bits as the final answer or at least as starting points for local search. $\endgroup$ Jun 29, 2016 at 12:46
  • $\begingroup$ I guess, that's why it is good to add context to the question when asking it, to avoid answers like mine :-) $\endgroup$
    – Suvrit
    Jun 29, 2016 at 13:07

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