Assume that we optimize a convex problem (convex objective and linear constraint) over a set of functions (say $L2$). Consider now the same optimization problem (same objective and same linear constraint) but now we optimize over a subset of $L2$. For example the set of affine functions or the set of polynomials of degree 2.

Is there a geometric relationship between the optimum of the first problem (optimizing over the set L2) and the second problem (optimizing over the subset of affine functions).

For example, can it be that the optimum of the second problem is the projection of the optimum of the first problem on the subset of affine functions in the sense of certain distance.

Does this question make sense and if so was this done in some textbook or perhaps an old paper? Many thanks for your help.