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Let $\mathcal P \subset \mathbf{R}^n$ be a polyhedron described by a set of linear inequalities, and $a$ a point in $\mathbf{R}^n$. Are the following problems easy or hard? (Easy means the solution can be found by solving one or a modest number of convex optimization problems.)

Find a point in $\mathcal P$ that is closest to $a$ in Euclidean norm.

Find a point in $\mathcal P$ that is closest to $a$ in $\ell_\infty$ norm.

Find a point in $\mathcal P$ that is farthest from $a$ in Euclidean norm.

Find a point in $\mathcal P$ that is farthest from $a$ in $\ell_\infty$ norm.