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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.

• Correct! This can be done by solving a QP.
• Incorrect.

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

• Correct! This can be done by solving an LP.
• Incorrect.

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

• Incorrect.
• Correct! This problem is NP-hard. In fact, it's hard to just check whether or not $\mathcal P$ lies inside a given Euclidean ball.

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

• Correct! This can be done by maximizing and minimizing each component of $x$, subject to $x \in \mathcal P$. This requires solving $2n$ LPs.
• Incorrect.