We define the polar dual of a polytope $P$ as the set $$\{x\in \mathbb{R}^n: x \cdot a\geq 1 \text{ for all } a\in P\}$$ Why do we require $1$ instead of $2$ or any other constant?
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It is just a normalization. If you were to pick any other constant (say, $c$, where $c$ is positive), then it would only scale the set by a factor of $1/c$. 

