We give three proofs of the fact that a smoothly bounded, convex domain in R n has defining functions whose Hessians are non-negative definite in a neighborhood of the boundary of the domain.

We introduce a polynomial time algorithm for optimizing the class of star-convex functions, under no Lipschitz or other smoothness assumptions whatsoever, and no restrictions except exponential boundedness on a region about the origin, and Lebesgue measurability. The algorithm’s performance is polynomial in the requested number of digits of accuracy and the dimension of the search domain. This ...