  If $f\in L^2(R^d)$ and if the function $f(x)f(y)$ is close in $L^2(R^{2d})$
norm to a radially symmetric function of $(x,y)$ then $f$ is close in $L^2$
norm to a centered Gaussian function. This is proved in a quantitative form
with the optimal exponent measuring closeness.
