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I wonder if there is any method to compute variational problems subject to certain shape constraints (e.g., convexity, monotonicity, etc.). The literature I found on this topic (which I am no expert in) seem to focus on existence and regularities of the solution.

For example: denote the collection of convex functions on $[-1,1]$ by $\mathcal{C}$. Given a fixed $f \in \mathcal{C}\cap L^2$, compute

$\sup\{|Lf-Lg|: \|f-g\|_2 \leq \epsilon, g \in \mathcal{C}\cap L^2\}$,

where $L$ is some linear functional. To be concrete, let's consider $Lf=f(0)$. When $\epsilon$ is small, $g$ is like a perturbation of $f$, but since $g$ needs to be convex, the perturbation cannot be arbitrary. Even some asymptotic result would be enlightening.

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for this particular problem, the solution g coincides with f outside some nbd of 0; locally at 0 it has a V-shaped graph, that touches the graph of f on both sides. –  Pietro Majer Nov 23 '11 at 7:25
    
Thanks Pietro! But it does not seem always true: take $f(x)=x$. The obtained $g$ is non-convex? For functions like $f(x)=x^2$, What you suggested should be optimal when $\epsilon$ is small: choose a $g$ composed of two tangent lines and the rest of the parabola. –  mr.gondolier Nov 23 '11 at 8:22
    
Of course: the graph of $f$ is inside the angle between the two half-lines of the prolongation of the V, so that $g$ is convex. This means that for ϵ small the V has to be quite open and $g$ remains close to $f.$ This does seem the case any ϵ>0 even large, and for $f(x)=x$ too; again the minimal g should be of the form $g(x)=ax^+ +bx^-+c$, for some $a,b,c$ –  Pietro Majer Nov 23 '11 at 18:42
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