The cancellation law for inf-convolution of convex functions

Dariusz Zagrodny

Studia Mathematica (1994)

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

Abstract

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Conditions under which the inf-convolution of f and g f □ g ( x ) : = i n f 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 l i m ∥ x ∥ → ∞ f ( x ) / ∥ x ∥ = ∞ 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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