If $X$ is a banach space, the Mazur's theorem shows that the norm and the weak closure of a convex set coincide.
The Mazur's theorem seems to be false for the weak-$*$ topology, that is, the norm and the weak-$*$ closure of a convex set can be different.
Do you know a simple counterexample showing this?
Every non-reflexive Banach space $X$ is a counterexample, as it is weak* dense in its second dual $X^{**}$. ($X$ is obviously convex, and is closed in the norm topology, since it is complete.) A more precise statement is that the closed unit ball $B_X$ of $X$ is weak* dense in the closed unit ball $B_{X^{**}}$ of $X^{**}$; this also shows that $B_X$ is an example.
A proof of the density claim can be found at The the image of the unit ball in X is weak-* dense in the unit ball of X**. A brief summary: if the claim is false, then some point $z\in B_{X^{**}}$ can be separated from $B_X$ by a linear functional coming from a vector $y\in X^*$. But this is impossible because if, say, $\langle y, x\rangle \le \alpha$ for all $x\in B_X$, then $\|y\|_{X^*} \le \alpha$, hence $\langle y, z\rangle \le \alpha$ for all $z\in B_{X^{**}}$.