Let \(A\) be an affinoid algebra over \(k\) and \(X=\mathrm{Sp}(A)\) be the associated affinoid space.
A subset \(R\subset X\) is rational if there are \((f_0,\dots,f_s)=A\) such that
\[R = \{x\in X \mid |f_i(x)|\le |f_0(x)|\text{ for all } i=1,\dots,s\}.\]
associated to \(R\) is the affinoid algebra (requiring convergence of \(f_i/f_0\))
\[B=A\langle Z_1,\dots,Z_s\rangle/(f_0Z_1-f_1,\dots,f_0Z_s-f_s).\]
Let \(\phi:A\to B_\{f_i\}\) be the obvious morphism of affinoid algebras induced by the above definition. Geometrically there is a canonical map
\[\mathrm{Sp}(\phi):\mathrm{Sp}(B)\to \mathrm{Sp}(A)=X\]
We have
\(\mathrm{Sp}(\phi)\mathrm{Sp}(B)\subset R\subset X\), the image of \(\mathrm{Sp}(B)\) lies inside \(R\).