By an ``image'' there is understood a real function g on an open rectangle \(R\subset {\mathbb{R}}^ 2\), the value g(x,y) being the ``grey- level'' at the point (x,y). An ``image-segmentation'' is a pair (u,B), where B is a finite set of piecewise \(C^ 1\) curves, called ``contours'', and u is a real function which is regular on connected components of \(R\setminus B\). In a ``good'' segmentation (u,B) the curves of B should be the boundaries of homogeneous areas in the image and u a sort of mean of g in the interior of such areas. The authors prove that the minimum of the functional \[ E(u,B):=\| u-g\|_{L^ 2(R)}+length(B) \] is attained at some B.