Remark 22.1.889911A537C86C

Remark 22.1.88.9911A537C86C Note that $\tau \mapsto A_{\tau }$ is upper semicontinuous, but not necessarily concave. Let $(A'_{\tau })_{\tau \in \mathbb {R}}$ be the upper concave envelope of $\tau \mapsto A_{\tau }$. Then it can be inductively constructed as follows:

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   For $\tau \in (\tau _0,\infty )$, we let $A'_{\tau }\equiv -\infty $;

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   we set $A'_{\tau _0}=A_{\tau _0}$;

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   define inductively for $j=0,\ldots ,N-1$ the following: For $\tau \in [\tau _{j+1},\tau _ j)$, we set

   \[ A'_{\tau }=\max _{i=j+1,\ldots ,N}\left( \frac{\tau _ j-\tau }{\tau _ j-\tau _ i}A_{\tau _ i}+\frac{\tau -\tau _ i}{\tau _ j-\tau _ i}A'_{\tau _ j} \right)\lor A'_{\tau _ j}; \]
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   for $\tau \in (-\infty ,\tau _ N)$, we set $A'_{\tau }=A_{\tau _ N}$.

This construction is just a reformulation of the general formula proposition 21.11.12.

In particular, $A'_{\tau }\in \mathrm{PSH}(X,\theta )$ for all $\tau \leq \tau _0$.

Note that $A'$ is not necessarily piecewise linear.