Define a map $\varphi \colon [0,1]\to C[0,1]^*$ by $\varphi(x) = \delta_x$. Then $\varphi$ is a homeomorphism for the w*-topology. Let $K$ denote the image of $\varphi$.
I have two questions:
1) Is the set $\overline{\mbox{conv}}^{\|\cdot\|}K$ compact for the weak* topology in $C[0,1]^*$ (in other words, is it weak*-closed)?
2) Is the set $\overline{\mbox{conv}}^{w^*}K$ compact in the weak topology of $C[0,1]^*$ (that is, the weak topology implemented by $C[0,1]^{**}$)?
Thank you.