Hamel Bases, Convexity and Analytic Sets in Fréchet Spaces

It is shown that a Hamel basis over the field of reals of an infinite dimensional linear Polish space can not be an analytic set. Furthermore, if $(x_{\alpha})$ is an infinite linearly independent subset of a Fr\'echet space $X$ and if $C$ is the convex cone generated by $(x_{\alpha}),$ then $C$ is not a closed set. In particular, the convex cone generated by a Hamel basis in such a space can not be closed.The notion of convex and midpoint convex functions extended to the case when the domain of the functions is a connected open set, and analytic graph theorems are given for these functions. It is shown also that if $f:{\mathbb R}^n \to {\mathbb R}$ is an order monotone function, then $f$ is Baire measurable, but in general, $f$ is not universally measurable