Chapter 3 Geometry of convex functions - Meboo Publishing ...

260 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS
Figure 77: Iconic unimodal differentiable quasiconvex function of two
variables graphed in R 2 × R on some open disc in R 2 . Note reversal of
curvature in direction of gradient.
3.9.0.0.1 Definition. Quasiconvex function.
f(X) : R p×k →R is a quasiconvex function of matrix X iff domf is a convex
set and for each and every Y,Z ∈domf , 0≤µ≤1
f(µY + (1 − µ)Z) ≤ max{f(Y ), f(Z)} (618)
A quasiconcave function is determined:
f(µY + (1 − µ)Z) ≥ min{f(Y ), f(Z)} (619)
Unlike convex functions, quasiconvex functions are not necessarily
continuous; e.g., quasiconcave rank(X) on S M + (2.9.2.6.2) and card(x)
on R M + . Although insufficient for convex functions, convexity of each and
every sublevel set serves as a definition of quasiconvexity:
△