Pseudo-convex

Kahler structures are easy to construct and flexible. For example, any complex submanifold of a Kahler manifold is again Kahler, and a Kahler metric is locally given by a Kahler potential, i.e. uj = / ddu for a strictly pseudo convex function u. However, hyper-Kahler structures are neither easy to construct nor flexible (even locally). A hypercomplex submanifold of a hyper-Kahler manifold must be totally geodesic, and there is no good notion of hyper-Kahler potential. The following quotient construction, which was introduced by Hitchin et al.[39] as an analogue of Marsden-Weinstein quotients for symplectic manifolds, is one of the most powerful tool for constructing new hyper-Kahler manifolds. 

This section presents the definitions, properties and relationships of quasi-convex, quasi-concave, pseudo-convex and pseudo-concave functions. 

Definition 2.3.5 (Pseudo-convex function) /(x) is pseudo-convex if for every xi,x2 S,... 

Properties of Pseudo-convex and Pseudo-concave Functions... 

Pseudo-convex and pseudo-concave functions exhibit the following properties ... 

Relationships among Convex, Quasi-convex and Pseudo-convex Functions... 

The relationships among convex, quasi-convex and pseudo-convex functions are summarized in the following ... 

Section 2.3 focuses on the generalizations of convex and concave functions and treats the quasi-convex, quasi-concave, pseudo-convex and pseudo-concave functions, and their properties. Further reading in this subject is the excellent book of Avriel et al. (1988). 

What additional conditions are needed in problem 19 so as to have pseudo-convexity in (i) and pseudo-concavity in (ii) ... 

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