# The cancellation law for inf-convolution of convex functions

Studia Mathematica (1994)

• Volume: 110, Issue: 3, page 271-282
• ISSN: 0039-3223

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## Abstract

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Conditions under which the inf-convolution of f and g $f\square g\left(x\right):=in{f}_{y+z=x}\left(f\left(y\right)+g\left(z\right)\right)$ 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 $f:X\to ℝ\cup +\infty$ on a reflexive Banach space such that $li{m}_{\parallel x\parallel \to \infty }f\left(x\right)/\parallel x\parallel =\infty$ constitutes a semigroup, with inf-convolution as multiplication, which can be embedded in the group of its quotients.

## How to cite

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Zagrodny, Dariusz. "The cancellation law for inf-convolution of convex functions." Studia Mathematica 110.3 (1994): 271-282. <http://eudml.org/doc/216114>.

@article{Zagrodny1994,
abstract = {Conditions under which the inf-convolution of f and g $f □ g(x):= inf_\{y+z=x\}(f(y)+g(z))$ 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 $f: X → ℝ ∪ \{+∞\}$ on a reflexive Banach space such that $lim_\{∥x∥ → ∞\} f(x)/∥x∥ = ∞$ constitutes a semigroup, with inf-convolution as multiplication, which can be embedded in the group of its quotients.},
author = {Zagrodny, Dariusz},
journal = {Studia Mathematica},
keywords = {inf-convolution; convex functions; subdifferentials; the cancellation law; a characterization of reflexivity; cancellation property; Banach space; semi-continuous convex functions; subdifferential},
language = {eng},
number = {3},
pages = {271-282},
title = {The cancellation law for inf-convolution of convex functions},
url = {http://eudml.org/doc/216114},
volume = {110},
year = {1994},
}

TY - JOUR
AU - Zagrodny, Dariusz
TI - The cancellation law for inf-convolution of convex functions
JO - Studia Mathematica
PY - 1994
VL - 110
IS - 3
SP - 271
EP - 282
AB - Conditions under which the inf-convolution of f and g $f □ g(x):= inf_{y+z=x}(f(y)+g(z))$ 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 $f: X → ℝ ∪ {+∞}$ on a reflexive Banach space such that $lim_{∥x∥ → ∞} f(x)/∥x∥ = ∞$ constitutes a semigroup, with inf-convolution as multiplication, which can be embedded in the group of its quotients.
LA - eng
KW - inf-convolution; convex functions; subdifferentials; the cancellation law; a characterization of reflexivity; cancellation property; Banach space; semi-continuous convex functions; subdifferential
UR - http://eudml.org/doc/216114
ER -

## References

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17. [17] D. Zagrodny, An example of bad convex function, J. Optim. Theory Appl. 70 (1991), 631-637. Zbl0793.90056

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