What is the relation between Chebyshev Norm and L1 norm in $D$ dimensions

Question

How do you translate one to the other (Chebyshev norm to the L1 norm) for $D$ dimensions?

Answer 1

You cannot convert one to the other. For some vectors in $d$-dimensions the Chebyshev norm (that is, the $\ell^\infty$ norm) and $\ell^1$ are identical. For other vector they can differ by up to a factor of $d$. The best you can say in $d$-dimensions is $$ \|x\|_\infty \le \|x\|_1 \le d \|x\|_\infty .$$