Appl Math Optim 45:169–184 (2002)
2002 Springer-Verlag New York Inc.
Permanently Going Back and Forth between the “Quadratic World”
and the “Convexity World” in Optimization
and M. Torki
Universit´e Paul Sabatier,
118 route de Narbonne, 31062 Toulouse Cedex, France
339 chemin des Meinajaries, 84911 Avignon, France
Abstract. The objective of this work is twofold. Firstly, we propose a review of
different results concerning convexity of images by quadratic mappings, putting
them in a chronological perspective. Next, we enlighten these results from a geo-
metrical point of view in order to provide new and comprehensive proofs and to
immerse them in a more general and abstract context.
Key Words. Quadratic functions, Range convexity, Joint positive deﬁniteness.
AMS Classiﬁcation. 90C20, 52A20, 15A60.
The “quadratic” character and the “convexity” one seem to belong to completely different
worlds in mathematics; although they are old and well known, quadratic mappings and
convex sets still continue to be objects of active research. It happens there are unexpected
but very interesting results of convexity concerning quadratic mappings. One of the goals
in this paper is to review the main ones, putting them in a chronological perspective
(Sections 1 and 2).
Section 1 deals with the convexity of images by quadratic mappings; we display
there results by Dines (1941), Brickman (1961), and Barvinok (1995), amongst others.
The ﬁrst results in this area seem due to Dines and Brickman: Dines showed that, for