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Fixed typo in Fenchel-Moreau.
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Bokuan Li
2026-06-24 19:48:55 -04:00
parent 32ba2483ad
commit 9b6385cc16
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src/fa/convex/legendre.tex
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@@ -214,7 +214,7 @@
so $(\Phi_t, \Gamma_t) \le f$. By (1), so $(\Phi_t, \Gamma_t) \le f$. By (1),
\[ \[
f^{**}(x) \ge \dpn{x, \Phi_t}{E} + \Gamma_t = \dpn{x, \phi_0}{E} + \gamma_0 + t\underbrace{\dpn{x, \phi}{E} - \gamma}_{> 0} f^{**}(x) \ge \dpn{x, \Phi_t}{E} + \Gamma_t = \dpn{x, \phi_0}{E} + \gamma_0 + t\underbrace{(\dpn{x, \phi}{E} - \gamma)}_{> 0}
\] \]
As the above holds for all $t > 0$, $f^{**}(x) = \infty \ge \alpha$. Since $f^{**}(x) \ge \alpha$ for all $(x, \alpha) \in E \times \real \setminus A$, $\text{epi}(f^{**}) \subset \ol{\text{Conv}}(\text{epi}(f))$. As the above holds for all $t > 0$, $f^{**}(x) = \infty \ge \alpha$. Since $f^{**}(x) \ge \alpha$ for all $(x, \alpha) \in E \times \real \setminus A$, $\text{epi}(f^{**}) \subset \ol{\text{Conv}}(\text{epi}(f))$.