The problem of approximating m data points $(x_i, y_i)$ in $R^n+1$, with a quadratic function $q(x,p)$ with s parameters $s<=m$, is considered. The parameter vector $p$ in $R^s$ is determined so as to satisfy three conditions: (1) $q(x,p)$ must underestimate all m data points, i.e. $q(x_i,p)<=y_i$, $i=1,....m$. (2) The error of approximation is to be minimized in the $L1$ norm. (3) The eigenvalues of $H$ are to satisfy specified lower and upper bounds, where $H$ is the Hessian of $q(x,p)$ with respect to $x$. Approximation of the data by a sum of negative Gaussians is also considered.