The free closed semialgebraic set Df determined by a hermitian noncommutative polynomial f∈Mδ(C<x,x∗>) is the closure of the connected component of {(X,X∗)∣f(X,X∗)≻0} containing the origin. When L is a hermitian monic linear pencil, the free closed semialgebraic set DL is the feasible set of the linear matrix inequality L(X,X∗)⪰0 and is known as a free spectrahedron. Evidently these are convex and it is well known that a free closed semialgebraic set is convex if and only it is a free spectrahedron. The main result of this paper solves the basic problem of determining those f for which Df is convex. The solution leads to an efficient algorithm that not only determines if Df is convex, but if so, produces a minimal hermitian monic pencil L such that Df=DL. Of independent interest is a subalgorithm based on a Nichtsingulärstellensatz presented here: given a linear pencil L˜ and a hermitian monic pencil L, it determines if L˜ takes invertible values on the interior of DL. Finally, it is shown that if Df is convex for an irreducible hermitian f∈C<x,x∗>, then f has degree at most two, and arises as the Schur complement of an L such that Df=DL.