# Minimizer of $p$-average of distances to points $x_1,\dots,x_n$

Let $(X,\|\cdot\|)$ be a normed space. Let $x_1,\dots,x_n$ be points in $X$. For a vector $x$ we define the distance vector $D(x) \in \mathbb R^n$ by

$D(x)_i = \| x - x_i \|$

Can you name the minimizer of the $p$-norm of $D$ for $1 \leq p \leq \infty$? Of particular interest is the case $X = \mathbb R$.

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so the number of points $x_1,...,x_n$ is the same as the dimension of $\mathbb{R}^n$? –  Paul Jul 11 '12 at 9:18
Are you just looking for a name? –  Christian Blatter Jul 11 '12 at 9:34
At least, when $X$ is a Hilbert space and $p=2$ we can compute it. –  Davide Giraudo Jul 11 '12 at 10:47
@Paul: Yes, by definition. The vector $D$ is just a formal device. –  Martin Jul 11 '12 at 12:37
@ChristianBlatter: A name, like some exotic mean, would be helpful, of course. –  Martin Jul 11 '12 at 12:38