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 Mar 30 comment Is this min not less than a min @domotorp, can you show that the maximum in the second quantity at the case that $v_{0},v_{1},v_{2},v_{3}$and $v_{4}$ are all on the boundary of $\mbox{convexhull}\left(v_{0},v_{1},v_{2},v_{3},v_{4}\right)$ is greater than or equal to the maximum in the second quantity at the case of regular pentagon? Thanks a lot! Mar 30 comment Is this min not less than a min @domotorp, Lagrange multipliers normally requires the function to be minimized to be differentiable, but the function here to be minimized is not differentiabe at many places. Mar 30 comment Is this min not less than a min @Günter, I calculated both the value for the square plus its center and the square plus an optimal point inside the square, and the values are both greater than the extremal value in the case when all the five points are on the boundary. But do you know if the square plus an point inside realizes the the minimum over the case of one point inside a quadrilateral, or how to find the optimal quadrilateral? Thanks! Feb 25 comment Is this min not less than a min domotorp , thanks for your answer, but the main difficulty in the question is how to show that the quantity for the case that one point inside a quadrilateral is greater than that for a regular pentagon. Do you know how to show that? Thanks! Feb 25 comment Is this min not less than a min Douglas, the maximum over $i, j, k$ is combinatorial, and it can probably be solved by using complex analysis techniques, since it seems difficult to solve by using other methods. Feb 25 revised Is this min not less than a min edited tags Feb 24 revised Is this min not less than a min edited tags; edited tags Feb 24 revised Is this min not less than a min edited body Feb 24 comment Is this min not less than a min Günter, yes, both of them are constant, since they are continous functions on compact sets. But do you how to find the minimizer and know why one of them is no less than the other? For a regular pentagon, it is $\frac{2 \left(\sqrt{7+2 \sqrt{5}}+4\right)}{\sqrt{5}}$, but how would you know that is a minimizer? Thanks. Feb 24 revised Is this min not less than a min edited tags Feb 24 comment Is this min not less than a min There had been no significant partitial results obtained. Any help that can lead to a partitial answer would also be appreciated. Feb 24 revised Is this min not less than a min edited tags Feb 24 revised Is this min not less than a min edited tags Feb 24 awarded Student Feb 24 revised Is this min not less than a min edited tags Feb 24 comment Is this min not less than a min or is there any counterexample? Thanks. Feb 24 revised Is this min not less than a min edited tags Feb 24 awarded Editor Feb 24 revised Is this min not less than a min added 1 characters in body Feb 24 comment Is this min not less than a min Thanks in advance for any correct answer or any help that can lead to a correct answer.