There are several notions of convexity at a point in the
literature, with applications to inequalities, the mechanics and
thermodynamics of continuous media, and nonlinear programming. We
came upon these notions in the process of constructing examples of
nonconvex minima, and we ended up introducing another one, inspired
by a definition of convexity with difference quotients. We study the
interrelationships of these notions for
real functions of one variable under various smoothness assumptions. To
argue that a “generic minimum” is nonconvex, we demonstrate how to
build one from any discontinuity of a second derivative.
