 Solution

Take $a=1$ and $b=k$, sois the capped simplex. A linear objective over this convex polytope attains its maximum at a zero-one extreme point. Choosing the $k$ coordinates at which $x_i$ is largest givesEquivalently, an exchange of weight from a smaller component to a larger one never decreases the objective. Thus the sum of the largest components is the support function ${\sigma }_C$.

Part c now givesBy the projection onto a box-constrained hyperplane, $v=P_C(y/t)$ hasConsequently the proximal operator is evaluated by solving this one-dimensional equation for $\nu$, then substituting the resulting projection.