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Nov 15, 2015 at 10:45 comment added Vectornaut @AlexanderChervov, note that you don't need to see the convex hull of $T = \{\sin(x + a) \colon a \in S^1\}$; you just need to find the unique constant function that can be written as a limit of elements of $T$. From the definition of the Riemann integral, I think you should be able to prove that $F(x) = \tfrac{1}{2\pi} \int_0^{2\pi} \sin(x + a)\;da$ is a limit of convex combinations of elements of $T$ in the uniform topology. As expected, $F(x) = 0$.
Jan 16, 2012 at 19:53 comment added Alexander Chervov Closure - in what topology ? If I take $G=S^1$ $f=sin(x)$, shifts are $sin(x+a)$... how to see what is convex hull ?
Jan 16, 2012 at 18:42 history answered Neil Strickland CC BY-SA 3.0