Deflection Conditions for the Strict Quasi-convexity of Sets

Abstract

We develop in this chapter sufficient conditions for a set $(D,\mathcal{P})$ to be s.q.c. As we have seen in Chap. 7, an s.q.c. set, which is characterized by the fact that R _G(D) > 0, has necessarily a finite curvature, as R(D) ≥ R _G(D) (see Proposition 7.2.9). But the condition

$$R(D)\ >\ 0$$

(8.1)

is not sufficient to ensure that $(D,\mathcal{P})$ is s.q.c.

This can be seen on the simple case where $(D,\mathcal{P})$ is an arc of circle of radius R and arc length L (Fig. 7.3). The deflection of this arc of circle, that is, the largest angle between two of its tangents – obviously the ones at its two ends in this case – is given by

$${\Theta }_{\mathrm{circle}}\ =\ L/R.$$

(8.2)