We examine the applications of cones whose elements are from linear functions spaces, and their applications in optimal geometric design. For instance, the set of symmetric matrix polynomials P(t) which are positive semidefinite for every value of t, or the set of vectors v(t) which belong to the second order cone for every of value of t, are known to be semidefinite representable. In addition, such sets arise in geometric problems. For instance in design of paths with constraints on the curvature, or in computing balls or ellipsoids of minimum volume containing a set of closes paths. We will discuss these connections and other related applications.