Abstract

We present two sufficient conditions in order that a real function on a finite-dimensional normed space be convex (Theorems 1 and 2) and show some consequences of them. In particular, it comes out that a real function ff on a finite-dimensional Hilbert space XX is convex, provided that ff has the property that for each point y∈Xy \in X and each λ>0\lambda > 0 the real function X∋x→λf(x)+∥x−y∥2X \ni x \to \lambda f(x) + \|x-y\|^2 has a unique global minimum

Author information

Contact details are reproduced from the original publication and may be historical.

Andrea Orazio Caruso

Dip. di Matematica e Informatica, Università di Catania, Viale A. Doria 6, 95125 Catania, Italy

aocaruso@dmi.unict.it

Alfonso Villani

Dip. di Matematica e Informatica, Università di Catania, Viale A. Doria 6, 95125 Catania, Italy

villani@dmi.unict.it

References

  1. F. Bernard, L. Thibault: Prox-regular functions in Hilbert spaces, J. Math. Anal. Appl. 303 (2005) 1–14.https://doi.org/10.1016/j.jmaa.2004.06.003Google Scholar
  2. G. Köthe: Topological Vector Spaces I, Springer, Berlin (1969).Google Scholar
  3. B. Ricceri: A conjecture implying the existence of non-convex Chebyshev sets in infinite-dimensional Hilbert spaces, Matematiche 65 (2011) 193–199.Google Scholar
  4. A. W. Roberts, D. E. Varberg: Convex Functions, Academic Press, New York (1973).Google Scholar
  5. X. Wang: On Chebyshev functions and Klee functions, J. Math. Anal. Appl. 368 (2010) 293–310.https://doi.org/10.1016/j.jmaa.2010.03.041Google Scholar

Suggested citation

A. O. Caruso, A. Villani. “Two Conditions for a Function to be Convex.” Journal of Convex Analysis 20 (2013), No. 4, 1189–1201. https://doi.org/10.68381/jca20064

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2013. Rights now held by Banach Press LLC.