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The cancellation law for inf-convolution of convex functions. (English) Zbl 0811.49012

The author obtains sufficient conditions for the following cancellation property: \[ f\square h\equiv g\square h\quad\text{implies}\quad f\equiv g, \] where \(f\square g(x)\) is defined by \(f\square g(x):= \inf_{y+ z= x}(f(y)+ g(z))\), called the inf-convolution of \(f\) and \(g\). One of the sufficient conditions established is that if \(X\) is a Banach space, \(f,g,h: X\to \mathbb{R}\cup \{+\infty\}\) are proper lower semi-continuous convex functions and \(h\) is also uniformly convex, then the cancellation property holds. The main tool used is the subdifferential analysis of convex functions. A counterexample of this cancellation property is given.

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
49J52 Nonsmooth analysis
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