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We will study an open problem pertaining to the uniqueness of minimizers fora class of variational problems emanating from Meyer's model for thedecomposition of an image into a geometric part and a texture part. Mainly, weare interested in the uniqueness of minimizers of the problem: \begin{equation}\label{eq:abs-1} (0.1)\qquad \inf \left\{J(u)+J^*\left(\frac{v}{\mu}\right):(u,v)\in L^2(\Omega)\times L^2( \Omega){,}\; f=u+v \right\} \end{equation}where the image $f$ is a square integrable function on the domain $\Omega$, thenumber $\mu$ is a parameter, the functional $J$ stands for the total variationand the functional $J^*$ is its Legendre transform. We will consider Problem(0.1) as a special case of the problem: \begin{equation} \label{eq:abs-2}(0.2)\qquad \inf\{s(f-u)+s^*(u): u\in \mathcal{X}\} \end{equation} where$\mathcal X$ is a Hilbert space containing $f$ and $s$ is a continuoussemi-norm on $\mathcal X$. In finite dimensions, we will prove that Problem(0.2) reduces to a projection problem onto the polar of the unit ballassociated to a given norm on an appropriate Euclidean space. We will alsoprovide a characterization for the uniqueness of minimizers for a more generalprojection problem defined by using any norm and any nonempty, closed, boundedand convex set of an Euclidean space. Finally, we will provide numericalevidence in favor of the uniqueness of minimizers for the decompositionproblem.

Uniqueness of Minimizers of Some Variational Problems Arising in Image Processing