PomeDefinition 19.6.7.7B20D9977149 Let $\Delta _{\bullet }\in \mathrm{TC}(\Delta )$. We define its Legendre transform as
\[ G[\Delta _{\bullet }] \colon \Delta \rightarrow [-\infty ,\infty ),\quad a\mapsto \sup \left\{ \tau <\Delta _{\max }:a\in \Delta _{\tau }\right\} . \]
Given a test function $G\colon \Delta \rightarrow [-\infty ,\infty )$, we define its inverse Legendre transform $\Delta [G]_{\bullet }$ as the Okounkov test curve relative to $\Delta $ defined as follows:
76C68DFB458D$\Delta [G]_{\max }=\sup _{\Delta } G$, and
41620508AE71for each $\tau <\sup _{\Delta } G$, we set
\[ \Delta [G]_{\tau }=\{ x\in \Delta :G\geq \tau \} . \]
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