Let $P$ be an $n$th degree polynomial. Find the $(n-1)$st degree polynomial $R$, whose arithmetic mean of distances to all of the points $P$ passes be minimum.

Best fit line using geometric distance (not vertical distance) is similar, but it doesn't help. Maybe it can be solved by using Deming Regression and limits ($\Delta x\to0$), but I don't know how to do it.

Calculus does what Deming Regression does, in a harder way. Maybe it's possible to use calculus "optimization" to do that, if it's not too complex.

  • 2
    $\begingroup$ I am confused with the term sum of distances to all of the points $P$ passes. In what space is this? In the real numbers this will always be infinitely large, since there are infinitely many points with distance $>0$? $\endgroup$
    – RoyPJ
    Oct 26, 2017 at 14:51
  • $\begingroup$ @RoyPJ see edit. $\endgroup$
    – MCCCS
    Oct 26, 2017 at 15:06


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