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Let $U\subset \mathbb R^2$ be a bounded open convex neighborhood of $0$. For every $\epsilon>0$ we have $\displaystyle \overline U=\bigcap_{H\supset U\atop H \text{ closed half-plane}} H \subset (1+\epsilon)U$, therefore $\displaystyle \bigcap_{H\supset U\atop H \text{ closed half-plane}} H \setminus (1+\epsilon)U=\emptyset.$ Hence, by compactness, the intersection is already empty on a finite sub-family of these half-planes $H_1,\dots, H_m$, so $\displaystyle U\subset \bigcap_{1\le j\le m} H_j \subset (1+\epsilon)U$, which exhibits a convex domain bounded by a polygon close to $U$.