# Semidefinite descriptions of the convex hull of rotation matrices

arXiv: https://arxiv.org/abs/1403.4914 · HTML: https://mlsanity.com/paper/1403.4914/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Mar 2014.

## Built on by (most cited first, 8 of 8)

- [Efficient Projection-Free Online Convex Optimization with Membership Oracle](https://mlsanity.com/paper/2111.05818/) (2021)
- [Advances in Global Solvers for 3D Vision](https://mlsanity.com/paper/2602.14662/) (2026)
- [Optimization and Sampling Under Continuous Symmetry: Examples and Lie Theory](https://mlsanity.com/paper/2109.01080/) (2021)
- [From Correspondences to Pose: Non-minimal Certifiably Optimal Relative Pose without Disambiguation](https://mlsanity.com/paper/2312.05995/) (2023)
- [Certifiably Optimal Anisotropic Rotation Averaging](https://mlsanity.com/paper/2503.07353/) (2025)
- [Correspondence-Free Fast and Robust Spherical Point Pattern Registration](https://mlsanity.com/paper/2508.02339/) (2025)
- [Fast, Robust, Permutation-and-Sign Invariant SO(3) Pattern Alignment](https://mlsanity.com/paper/2512.00659/) (2025)
- [Orientation Estimation of Cryo-EM Images for Molecules with Arbitrary Symmetry by Semidefinite Programming](https://mlsanity.com/paper/2610.18691/) (2026)

## Abstract (arXiv)

We study the convex hull of SO(n), thought of as the set of n× n orthogonal matrices with unit determinant, from the point of view of semidefinite programming. We show that the convex hull of SO(n) is doubly spectrahedral, i.e. both it and its polar have a description as the intersection of a cone of positive semidefinite matrices with an affine subspace. Our spectrahedral representations are explicit, and are of minimum size, in the sense that there are no smaller spectrahedral representations of these convex bodies.