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Geometrical problems
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.
Easy.
Correct!
This can be done by solving a QP.
Hard.
Incorrect.
Find a point in $\mathcal P$ that is closest to $a$ in $\ell_\infty$ norm.
Easy.
Correct!
This can be done by solving an LP.
Hard.
Incorrect.
Find a point in $\mathcal P$ that is farthest from $a$ in Euclidean norm.
Easy.
Incorrect.
Hard.
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.
Easy.
Correct!
This can be done by maximizing and minimizing each component of $x$, subject to $x \in \mathcal P$. This requires solving $2n$ LPs.
Hard.
Incorrect.