Definition of Gelfand width

Question

I'm reading an article where the Gelfand width is introduced as follows:

Given a compact set $K$ in the Banach space $X$, it's Gelfand width is defined by

$d^n(K)_X:=\inf\limits_{codim(Y)=n}\:\sup\limits_{x \in K\cap Y} ||x||_X$.

Does someone have an intuitive explanation or an example for this definition?