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In this chapter, we define a new class of subsets of a Hilbert space, called the quasi-convex sets to which properties (i) (uniqueness), (iii) (stability) and (iv) (existence as soon as the set is closed) of Proposition 4.1.1 can be generalized, provided they are required to hold only on some neighborhood. Technically, the whole chapter will consist in adapting the classical proofs for convex sets to the case where the segments are replaced by paths with finite curvature.
We postpone to Chap. 7 the generalization of property (ii) on the absence of parasitic stationary points.