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When doing

Plot[ArgMin[Norm[x - {-3, -1, 0, 5}, p], x], {p, 1, 3}]

I get this error message:

Plot::exclul: {(Abs[-5+x]^p+Abs[x]^p+Abs[1+x]^p+Abs[3+x]^p)-0,(3+x)-0,(1+x)-0,x-0,(-5+x)-0,
 p-0,Im[Abs[-5+x]^p+Abs[x]^p+Abs[1+x]^p+Abs[3+x]^p]-0}
 must be a list of equalities or real-valued functions.

The plot itself seems to be OK

enter image description here

-- although I am not sure because of this error. What caused it?

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This is a bug that seems to have been resolved in version 11.1, and is related to the exclusion detection functionality used by Mathematica to detect discontinuities. (The presence of Abs[] in the function being analyzed is what triggered the exclusion detection.)

Since you know that the function is continuous in your range of interest, just add Exclusions -> None:

Plot[ArgMin[Norm[x - {-3, -1, 0, 5}, p], x], {p, 1, 3}, 
     Exclusions -> None]
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