I have a problem involving recurrence and euclidean distances in $n$-dimensional curves.
Given the sequence of curves in $\mathbb{R}^n:$
$\{ x_{1}^2+x_{2}^2+\cdots + x_{n}^2 = 1, x_{1}^4+x_{2}^4+\cdots + x_{n}^4 = 1, \ldots, x_{1}^{2k}+x_{2}^{2k}+\cdots + x_{n}^{2k} = 1, \ldots \},$
and $d_{k}$ the maximum distance between the origin $(0,0,\ldots,0)$ and each curve.
Does exist a recurrence relation $d_k = f(d_{k-1})$ for all $k > 1$?
$d_1=1$as that is a sphere. For higher exponents, the maximum distance comes when all the variables are equal, when all the $x_i=r_m$ with $nr_m^{2m}=1$. Then $r_m=\sqrt[2m]{\frac 1n}=n^{\frac {-1}{2m}}$ and the distance is $d_m=\sqrt{nr_m^2}=n^{\frac 12 -\frac{1}{2m}}$