Suppose that $q$ is a quadratic form on $\mathbb{R}^n$, $q(x)=(x,Ax)$ say (or $q(x)=x^TAx$ if you prefer that notation). Then one could consider the quantity $$ \sup\{ \left|q(x)\right| : \left\| x \right\| \leq 1 \}. $$ Is this an interesting quantity? In particular when the norm is the $p$-norm for $p \neq 2$?
norm of a quadratic form
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2Note that $\sup \{ |q(x)|: \| x \| \le 1 \} = \sup \{ |q(x): \| x \| =1 \}$ by applying linearity: $q(\lambda x)=\lambda^2 q(X)$, so $q(x) = \|x\|^2 q(x/\|x\|)$. – 2012-05-03