Convex set for similarity constraint

[Squatchmichae]

I am trying to ultimately find the projector onto a convex set defined in a non-explicit way, for a seismic processing application.

The signals in question are members of some Hilbert Space H and the set membership requires that they must correlate with each other above some scalar $\rho$, given that the known signal $\textbf{w}$ is in the set. Symbolically, I want to find a projector $\textit{P}$ onto the convex set $\textit{C}$:

\begin{equation}

C = \left\{\mathbf{u}(t) : \left\langle \hat{\mathbf{u}}(t),\hat{\mathbf{v}}(t) \right\rangle \geq\rho_{0}, \forall \mathbf{v}(t) \in C, \quad where \quad \mathbf{w}(t) \in C \right\},

\end{equation}

Any intermediate help is appreciated, i.e., is there an equivalent way to formulate this set, that make finding the projector easier?

 

[Office_Shredder]

I'm a bit confused by how C references itself in its definition.

 

[Squatchmichae]

I'm a bit confused by how C references itself in its definition.

I understand the confusion--that is what makes the defining characteristic a little awkward. The basic idea is this: each element in the convex set $\textit{C}$ must correlate with every other element above $\rho$. But we also know that a given (known) element $\textbf{w}(t)$ is contained in $\textit{C}$. Is that less confusing of a statement?