# Polar duality and -1 [closed]

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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Any other negative constant works. Other formulas have a normalizing constant that will change if you use something other than $-1$. The value $-1$ is used because it is the "simplest" negative number and for that choice, the dual of a unit round ball (admittedly, not a polytope) is a unit round ball. –  Deane Yang Aug 30 '12 at 18:06
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 $1/c$.