70 research outputs found
Algebraic Boundaries of Convex Semi-algebraic Sets
We study the algebraic boundary of a convex semi-algebraic set via duality in
convex and algebraic geometry. We generalize the correspondence of facets of a
polytope to the vertices of the dual polytope to general semi-algebraic convex
bodies. In the general setup, exceptional families of extreme points might
exist and we characterize them semi-algebraically. We also give an algorithm to
compute a complete list of exceptional families, given the algebraic boundary
of the dual convex set.Comment: 13 pages, 2 figures; Comments welcom
Generic Spectrahedral Shadows
Spectrahedral shadows are projections of linear sections of the cone of
positive semidefinite matrices. We characterize the polynomials that vanish on
the boundaries of these convex sets when both the section and the projection
are generic.Comment: version to be publishe
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