Abstract
We study the algebraic boundary of a convex semi-algebraic set via duality in convex and algebraic geometry. We generalise the correspondence of facets of a polytope with the vertices of the dual polytope to general semi-algebraic convex sets. In this case, exceptional families of extreme points might exist and we characterise them semi-algebraically. We also give a strategy for computing a complete list of exceptional families, given the algebraic boundary of the dual convex set.
Highlights
The algebraic boundary of a semi-algebraic set is the smallest algebraic variety containing its boundary in the Euclidean topology
The algebraic boundary of a convex set which is not a polytope has recently been considered in other special cases, most notably the convex hull of a variety by Ranestad and Sturmfels, cf. [11] and [12]
This article is organised as follows: In Section ‘The algebraic boundary and convexity’, we introduce the algebraic boundary of a semi-algebraic set and discuss some special features of convex semi-algebraic sets coming from their algebraic boundary
Summary
The algebraic boundary of a semi-algebraic set is the smallest algebraic variety containing its boundary in the Euclidean topology. If Y is an irreducible component of the algebraic boundary of C∨, the dual variety to PY is an irreducible subvariety of PExra(C), the set (PY )∗ ∩ Exr(C) is Zariski dense in (PY )∗ and the above condition on the normal cone is satisfied at a general extreme ray for the affine cone over (PY )∗.
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