Equivalence between p-cyclic quasimonotonicity and p-cyclic monotonicity of affine maps

Abstract

We prove that the notions of -cyclic quasimonotonicity and -cyclic monotonicity are equivalent for affine maps defined on Banach spaces. First this is done in a finite dimensional space by using the index of asymmetry for matrices defined by J.-P. Crouzeix and C. Gutan. Then this equivalence is extended to general Banach spaces.

Keywords:

• affine multivalued maps
• positive semidefinite matrices
• cyclic monotonicity
• monotonicity⁺
• paramonotonicity
• cyclic quasimonotonicity
• index of asymmetry

AMS Subject Classifications:

• 15A48
• 54C60

Acknowledgments

The authors thank the corresponding Editor and two anonymous referees for their helpful suggestions.

Notes

Dedicated to Jean-Pierre Crouzeix on the occasion of his 70th birthday.