EDB — 13P

Exercises

1. [13P]Topics:inf-convolution. Difficulty:*. When \((X,d)\) is a metric space, and \(f:X→ℝ∪\{ +∞\} \) is l.s.c. and bounded from below, let

   \[ f_ n(x) {\stackrel{.}{=}}\inf _{y∈ X} \{ f(y) + n d(x,y) \} \]

   be the inf-convolution. Show that the sequence \(f_ n\) is an increasing sequence of Lipschitz functions with \(f_ n(x)→_ n f(x)\).

   Solution 1

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