Given a set of $k-\text{dimensional}$ points $S$, how do you find the point $p$ that minimizes the sum of the distances from $p$ to each point in $S?$
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So you mean to minimize the sum $$ \sum_{i=1}^{\text{number of points}} \sqrt{\sum_{j=1}^{\text{dimensions}} (\xi_j-x_{ij})^2} $$ for a given set of points{$x_i$} , where the $i$th of them has coordinates $x_{ij}$? – Physor Sep 14 '20 at 14:29
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Yes. I suppose you would be trying to find the value that would minimize that sum. – Gilad Felsen Sep 14 '20 at 14:32
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denote the sum as $f(\xi_1,...,\xi_k)$, since it is a multivariable function. Now a necessary condition is that all partial derivatives must vanish simultaneously, but that is not sufficient – Physor Sep 14 '20 at 14:36
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2https://en.wikipedia.org/wiki/Geometric_median – RobPratt Sep 14 '20 at 15:07