  The Lovasz extension of a pseudo-Boolean function $f : \{0,1\}^n \to R$ is
defined on each simplex of the standard triangulation of $[0,1]^n$ as the
unique affine function $\hat f : [0,1]^n \to R$ that interpolates $f$ at the
$n+1$ vertices of the simplex. Its degree is that of the unique multilinear
polynomial that expresses $f$. In this paper we investigate the least squares
approximation problem of an arbitrary Lovasz extension $\hat f$ by Lovasz
extensions of (at most) a specified degree. We derive explicit expressions of
these approximations. The corresponding approximation problem for
pseudo-Boolean functions was investigated by Hammer and Holzman (1992) and then
solved explicitly by Grabisch, Marichal, and Roubens (2000), giving rise to an
alternative definition of Banzhaf interaction index. Similarly we introduce a
new interaction index from approximations of $\hat f$ and we present some of
its properties. It turns out that its corresponding power index identifies with
the power index introduced by Grabisch and Labreuche (2001).
