Plaumann, DanielSinn, RainerWesner, Jannik Lennart2023-06-122023-06-122022-08-06http://hdl.handle.net/2003/41731http://dx.doi.org/10.17877/DE290R-23574We study families of faces for convex semi-algebraic sets via the normal cycle which is a semi-algebraic set similar to the conormal variety in projective duality theory. We propose a convex algebraic notion of a patch—a term recently coined by Ciripoi, Kaihnsa, Löhne, and Sturmfels as a tool for approximating the convex hull of a semi-algebraic set. We discuss geometric consequences, both for the semi-algebraic and convex geometry of the families of faces, as well as variations of our definition and their consequences.enSemi-algebraic setsConvex setsNormal cycleFamilies of faces510Text