Permanently Going Back and Forth between the ``Quadratic World'' and the ``Convexity World'' in Optimization

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 geometrical point of view in order to provide new and comprehensive proofs and to immerse them in a more general and abstract context. \keywords{Quadratic functions, Range convexity, Joint positive definiteness.} \amsclass{90C20, 52A20, 15A60.}