Is there some notation and name for this norm? Given a field $K$ with an absolute value (you may imagine $\mathbb{R},\mathbb{C}$ or a $p$-adic field), I wonder if there is some notation and name (like $\|\|_\infty$ for the infinite [or supremum, or maximum] norm, $\|\|_2$ for the Euclidean norm or $\|\|_1$ for the Manhattan [or taxi] norm etc.) for the norm given by
$$
\|(z_0,\dots,z_d)\|=\max_{i,j}\{|z_i-z_j|\}(=\max_i\{z_i\}-\min_j\{z_j\}\text{, when }K=\mathbb{R})
$$
 A: The closest thing I know that has the same behavior is the diameter of a subset for a given distance. 
It is defined by : 
$$
\operatorname{diam}(A) = \sup \{ d(x, y) | x, y ∈ A \} 
$$
With $d$ the distance on your metric space. The diameter coincides with your definition when using real numbers, $d: A^2 \to \mathbb R; (x,y) \mapsto |x - y|$ where $A = \{ z_0, ..., z_d \} \subset \mathbb R$ is a finite set.
See https://en.wikipedia.org/wiki/Diameter#Generalizations for details
A: Since this norm on $K^{d+1}$ factorizes by $K\cdot(1,\dots,1)$, it seems natural call it the tropical norm and denote it by $\|\|_{trop}$. After think this and a small search on the web, I have found a paper with the same criterious (http://arxiv.org/pdf/1505.02045.pdf) and also a book which seems to apply also it (up to a page -410- which does not appear): https://books.google.es/books?id=iibCrkwJ2ysC&pg=PA414&lpg=PA414&dq=%22tropical+norm%22&source=bl&ots=G9BeKX5eC-&sig=ejBSlpj0f0yKqGBr3JQ7nMUt2Pg&hl=gl&sa=X&ved=0ahUKEwje9LTK1LLPAhVD7hoKHZymAk8Q6AEIKzAD#v=onepage&q=%22tropical%20norm%22&f=false
