The semialgebraic set Df determined by a noncommutative polynomial f is the closure of the connected component of {(X,X∗):f(X,X∗)≻0} containing the origin. When L is a linear pencil, the 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 by a theorem of Helton \& McCullough, a free semialgebraic set is convex if and only it is a free spectrahedron. \\\\ In this talk we solve the basic problem of determining those f for which Df is convex. The solution leads to an effective algorithm that not only determines if Df is convex, but if so, produces a minimal linear pencil L such that Df=DL. Of independent interest is a subalgorithm based on a Nichtsingulärstellensatz: given linear pencils L,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 noncommutative polynomial, then f has degree at most two, and arises as the Schur complement of an L such that Df=DL.