It is known (even by me) that the Chebyshev polynomial of degree $n$ (of the first kind) is the minimal polynomial in the space $L^{\infty}([-1,1])$ for a fixed $n$ and leading coefficient $2^n$.
However, what are the minimal polynomials for the $p$-norm in general for a fixed $n$? Does there exist a general answer?
This is my first question here and I apologize if it is not up to par. Feel free to edit, migrate or close it if necessary.