Seshadri-type constants and Newton-Okounkov bodies for non-positive at infinity valuations of Hirzebruch surfaces

We consider flags $E_{\bullet }=\{X\supset E\supset \{q\}\}$, where $E$ is an exceptional divisor defining a non-positive at infinity divisorial valuation ${\nu }_E$ of a Hirzebruch surface ${\mathbb{F}}_{\delta }$ and $X$ the surface given by ${\nu }_E,$ and determine an analogue of the Seshadri constant for pairs $({\nu }_E,D)$, $D$ being a big divisor on ${\mathbb{F}}_{\delta }$. The main result is an explicit computation of the vertices of the Newton-Okounkov bodies of pairs $(E_{\bullet },D)$ as above, showing that they are quadrilaterals or triangles and distinguishing one case from another.