Let $K$ be a convex,balanced and compact subset of a Banach space $X$. We let $X_{K}:=span\{K\}$. Define the norm $\x\_{K}:=\inf\{\alpha>0: x\in \alpha K\}, x\in X_{K}$. Then $(X_{K},\\_{K})$ is a Banach space and the unit ball of $(X_{K},\\_{K})$ is $K$. My question is : Is the space $(X_{K},\\_{K})$ reflexive? The answer maybe has already been known, but I do not know.

No it is not reflexive in general. Your assumptions basically say that $X_K$ is compactly embedded in $X$ because $K$ is more or less the unit ball of $X_K$ (it is equal to the closed unit ball I think). Now there are obviously a ton of examples of nonreflexive Banach spaces that are compactly embedded into other (completely arbitrary!) Banach spaces like the Sobolev space $W^{1,\infty}(0,1)$ (=the space of Lipschitz continuous functions) which compactly embeddeds into $C^0[0,1]$. 


Even if $K$ is just weakly compact, $X_K$ is isometrically isomorphic to a dual space with the weak$^*$ topology agreeing with the weak topology from $X$ on $K$. You can get a weakly compact convex symmetric set $J\supset K$ s.t. $X_J$ is reflexive. See my paper with Davis, Figiel, and Pelczynski, ``Factoring weakly compact operators". 

