The Hyperrigidity Conjecture for compact convex sets in ${\mathbb{R}}^2$

We prove that for every compact, convex subset $K\subset {\mathbb{R}}^2$ the operator system $A(K)$, consisting of all continuous affine functions on $K$, is hyperrigid in the C*-algebra $C(\mathrm{e}\mathrm{x}(K))$. In particular, this result implies that the weak and strong operator topologies coincide on the set $$\{T\in \mathcal{B}(H);T\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{d}\sigma (T)\subset \mathrm{e}\mathrm{x}(K)\}\mathrm{.}$$ Our approach relies on geometric properties of $K$ and generalizes previous results by Brown.