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We define $f:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}$ by $f(u,v)=\displaystyle\sum_{i=1}^n|u_i-v_i|^p$.
We'd like to minimize $f$ under the following constraints: $$ \left\{\begin{array}{l} g_1(u,v):=f(u,0)=1 \\ g_2(u,v):=f(0,v)=1 \\ g_3(u,v):=\langle u,v\rangle=0\end{array}\right. . $$ I know one way is to use Langrage multipliers to find the local maximum and minimum, so I restricted $f$ to the set $$D=\{(u_1,...,u_n,v_1,...,v_n):u_i\neq0,v_i\neq0,u_i\neq v_i,\forall 1\leq i\leq n\},$$ tried to solve the system of equations given by $$ \nabla f=\lambda_1\nabla g_1 + \lambda_2\nabla g_2 + \lambda_3\nabla g_3, $$ and came up with no conclusion.

Am I following the right path? Is there other efficient way to minimize $f$? Any help with solving the system of equations?

Any help would be highly appreciated.

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  • $\begingroup$ Do you know anything about $p$? Is it greater than $1$? This would make the problem differentiable at least $\endgroup$ – RobertHannah89 Nov 28 '16 at 17:01
  • $\begingroup$ Yes, here I'm supposing p greater than 1 $\endgroup$ – André Porto Nov 29 '16 at 1:13

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