Suppose $E$ is a Banach space.
Motivation: If $f \in E^*$, then the numerical values of $f$ have a simple geometric meaning,
$$|f(x)| = |f|\cdot d(x,\ker(f))$$
Question:
If $B: E \times E \rightarrow \mathbb{R}$ is a continuous bilinear form, in what ways can $B(x,x)$ be interpreted geometrically as a (function of) distance to $\ker(B)$, in $E$?
It seems like this depends heavily on the bilinear form, even in low dimensions.
Note that $B$ can be viewed as an element of the dual of the projective product $E \hat{\otimes}_{\pi}E$, so $B(x,x) = |B| \cdot d(x \otimes x, \ker(B))$.