The cancellation law for inf-convolution of convex functions

Conditions under which the inf-convolution of f and g ${\displaystyle f\square g\left(x\right):=\in f_{y+z=x}(f\left(y\right)+g\left(z\right))}$ has the cancellation property (i.e. f □ h ≡ g □ h implies f ≡ g) are treated in a convex analysis framework. In particular, we show that the set of strictly convex lower semicontinuous functions ${\displaystyle f:X\to ℝ\cup \{+\infty \}}$ on a reflexive Banach space such that ${\displaystyle \underset{\parallel x\parallel \to \infty }{lim}\frac{f\left(x\right)}{\parallel }x\parallel =\infty }$ constitutes a semigroup, with inf-convolution as multiplication, which can be embedded in the group of its quotients.