Algebraic Boundaries of $$\mathrm{SO}(2)$$ SO ( 2 ) -Orbitopes

Abstract

Let $$X\subset \mathbb{A }^{2r}$$XźA2r be a real curve embedded into an even-dimensional affine space. We characterise when the $$r$$rth secant variety to $$X$$X is an irreducible component of the algebraic boundary of the convex hull of the real points $$X(\mathbb{R })$$X(R) of $$X$$X. This fact is then applied to $$4$$4-dimensional $$\mathrm{SO}(2)$$SO(2)-orbitopes and to the so called Barvinok---Novik orbitopes to study when they are basic closed semi-algebraic sets. In the case of $$4$$4-dimensional $$\mathrm{SO}(2)$$SO(2)-orbitopes, we find all irreducible components of their algebraic boundary.