A local version of Karamardian’s theorem on lower semicontinuous strictly quasiconvex functions

It is known that lower semicontinuous strictly quasiconvex functions are quasiconvex. It is shown that ‘semilocally’ strictly quasiconvex functions are ‘semilocally’ quasiconvex on a ‘locally star shaped’ domain if some condition of ‘semilocal lower semicontinuity’ is satisfied. These things are analyzed in some detail. The functions involved have a connected quasiorder as their range. As a motivation, it is shown that a multiobjective function is quasiconvex and strictly quasiconvex with respect to the lexminmax order if its components have the same properties. There is a discussion of some applications to multiobjective decision, particularly with lexminmax as the underlying decision criterion.