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We define a convex-polynomial to be one that is a convex combination of themonomials $\{1, z, z^2, \ldots\}$. This paper explores the intimate connectionbetween peaking convex-polynomials, interpolating convex-polynomials, invariantconvex sets, and the dynamics of matrices. In particular, we use theseintertwined relations to both prove which matrices are convex-cyclic while atthe same time proving that we can prescribe the values and a finite number ofthe derivatives of a convex-polynomial subject to certain natural constraints.These properties are also equivalent to determining those matrices whoseinvariant closed convex sets are all invariant subspaces. Our characterizationof the convex-cyclic matrices gives a new and correct proof of a similar resultby Rezaei that was stated and proven incorrectly.

Convex-Cyclic Matrices, Convex-Polynomial Interpolation & Invariant Convex Sets