Robust stability of convex and compact sets of complex polynomials

A necessary and sufficient condition for a convex and compact set of complex polynomials to be strictly Hurwitz is given. The result implies and generalizes several results on strict Hurwitz property of polynomials. In particular, our result covers the "edge theorem" for polytopes of strictly Hurwitz polynomials and requires weaker conditions than the "edge theorem". It is also shown that previously established results on the robust strict Hurwitzness of diamond polynomials can be implied by our result. Finally, applying the result, we derive a new result on the robust positivity of a complex diamond rational function.