= Nonconvexity of Mumford–Shah segmentation
On the unit square, let $g=0$ and take $u_0=\chi_{\{x_1>0.4\}}$, $u_1=\chi_{\{x_1>0.6\}}$. Both have one unit-length jump and zero ordinary <gradient>. Their mean has two unit-length jumps. For squared-error coefficient one, $[E(u_0)+E(u_1)]/2=0.5+\beta$ while $E((u_0+u_1)/2)=0.45+2\beta$, violating <convexity> when $\beta>0.05$. The <Mumford–Shah functional> can therefore be nonconvex even though the fixed-edge reconstruction subproblem is <strictly convex>.
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