85 research outputs found
Orbitopes
An orbitope is the convex hull of an orbit of a compact group acting linearly
on a vector space. These highly symmetric convex bodies lie at the crossroads
of several fields, in particular convex geometry, optimization, and algebraic
geometry. We present a self-contained theory of orbitopes, with particular
emphasis on instances arising from the groups SO(n) and O(n). These include
Schur-Horn orbitopes, tautological orbitopes, Caratheodory orbitopes, Veronese
orbitopes and Grassmann orbitopes. We study their face lattices, their
algebraic boundary hypersurfaces, and representations as spectrahedra or
projected spectrahedra.Comment: 37 pages. minor revisions of origina
Semidefinite descriptions of the convex hull of rotation matrices
We study the convex hull of , thought of as the set of
orthogonal matrices with unit determinant, from the point of view of
semidefinite programming. We show that the convex hull of 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.Comment: 29 pages, 1 figur
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