diff --git a/src/fa/convex/legendre.tex b/src/fa/convex/legendre.tex index 02cd050..536a694 100644 --- a/src/fa/convex/legendre.tex +++ b/src/fa/convex/legendre.tex @@ -214,7 +214,7 @@ 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 $(x, \alpha) \in E \times \real \setminus A$, $\text{epi}(f^{**}) \subset \ol{\text{Conv}}(\text{epi}(f))$.