OK, there's more to it but I couldn't fit everything in the Title.
This is the situation: I have a subspace of $R^n$, call it $I$, which is contained in $Z \equiv R^n \cap \{x : \sum_i x_i = 0 \}$ and a vector $v \in Z\setminus I$. I know that the orthogonal (Euclidean inner product) projection of $v$ onto $I$ (call it $v_I$) minimizes the norm of $v - y$ over all $y \in I$, but I am interested in finding a vector $v^\prime \in I$ such that the maximum norm of $v - y$, as $y$ varies over $I$, is minimized at $v^\prime$.
I would venture to say that, in general, $v_I \neq v^\prime$, but there are additional restrictions on $I$ so I'd like to investigate conditions under which in fact $v_I$ is equal to $v ^\prime$ (i.e. $v_I$ minimizes, over all $y \in I$, both the Euclidean norm and the maximum norm of $v - y$).
Thanks for any information or pointers to resources.