Quasi-convex functions on subspaces and boundaries of quasi-convex sets

Abstract

We embed truncations of the epi-graph of quasi-convex functions defined on linear subspaces E ⊂ M^{N × n} of real matrices into M^{N × n} to bound quasi-convex sets by the graph of the functions. We also characterize subspaces E on which all quasi-convex functions are convex and show, by using the Tarski–Seidenberg theorem in real algebraic geometry, that if dim (E) > N + n − 1, then there exist non-trivial quasi-convex functions on E.