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Consider the function $g: \mathbb R^n \rightarrow \mathbb R$ given by: $$ g(x) = \arg\min_{y\in\mathbb R} \sum_{i=1}^n f_i(|y - x_i|) $$ where $f_i$ are convex, strictly increasing and continuous. Further assume that at least one $f_i$ is strictly convex (so the minimizer is unique). Is $g$ a continuous function?

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  • $\begingroup$ I would start with the case $n=1$. How would you treat the situation in which the functions $f_i$ do not have unique minimums? $\endgroup$ – hardmath Jan 12 '16 at 3:54
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    $\begingroup$ Just because the functions $f$ are convex does not mean their composition with absolute values will be. $\endgroup$ – Michael Grant Jan 12 '16 at 6:20
  • $\begingroup$ Updated the question with more conditions on $f_i$'s. Thanks for pointing this out. $\endgroup$ – ppd Jan 12 '16 at 19:43
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As you've written it, $g(x)$ appears to be a set valued function: the minimum can occur at many values of $y$. For instance, when each $f_i$ is constant we have $f(x)=\mathbb R$ for any value of $x$.

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  • $\begingroup$ Updated the question with more conditions on $f_i$'s. Thanks for pointing this out. $\endgroup$ – ppd Jan 12 '16 at 19:34
  • $\begingroup$ This should be a comment, not an answer... $\endgroup$ – dohmatob Jan 13 '16 at 4:15

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