Abstract. A connected, linearly ordered path γ ⊂ R n satisfying x 1 , x 2 , x 3 ∈ γ, and x 1 ≺ x 2 ≺ x 3 =⇒ |x 2 − x 1 | ≤ |x 3 − x 1 | is shown to be a rectifiable curve; a priori bounds for its length are given; moreover, these paths are generalized steepest descent curves of suitable quasi convex functions. Properties of quasi convex families are considered; special curves related to quasi convex families are defined and studied; they are generalizations of steepest descent curves for quasi convex functions and satisfy the previous property. Existence, uniqueness, stability results and length's bounds are proved for them.Résumé. Nous démontrons que les chemins γ ⊂ R n qui sont connectés et ordonnés, avec la proprieté de monotonicitésont des courbes. Des limitations pour leur longueur sont prouvé. Ces chemins sont des généralisations de courbes de la plus grande pente pour appropriées fonctions quasiconvexes. Propriétés des familles quasiconvexes et courbes liées avec elles sontétudiées. Nous démontrons l'existence, l'unicité, la dépendance continue de ces courbes avec des limitations pour leur longueur.2000 Mathematics Subject Classifications. Primary 52A20; Secondary 52A10, 52A38, 49J53.

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Two dimensional steepest descent curves (SDC) for a quasi convex family are considered; the problem of their extensions (with constraints) outside of a convex body K is studied. It is shown that possible extensions are constrained to lie inside of suitable bounding regions depending on K. These regions are bounded by arcs of involutes of ∂K and satisfy many inclusions properties. The involutes of the boundary of an arbitrary plane convex body are defined and written by their support function. Extensions SDC of minimal length are constructed. Self contracting sets (with opposite orientation) are considered, necessary and/or sufficients conditions for them to be subsets of a SDC are proved.

Let C be the family of compact convex subsets S of the hemisphere in R n with the property that S contains its dual S * ; let u ∈ S * , and let (S, u) = 2 ω n S θ, u dσ (θ). The problem to study inf (S, u), S ∈ C, u ∈ S * is considered. It is proved that there exists a minimizing couple (S, u) such that S is self-dual and u is on its boundary. More can be said for n = 3: the minimum set is a Reuleaux triangle on the sphere. The previous problem is related to the one to find the maximal length of steepest descent curves for quasi-convex functions, satisfying suitable constraints. For n = 2, let us refer to [Manselli P, Pucci C. Maximum length of steepest descent curves for quasi-convex functions. Geom. Dedicata. 1991]. Here, quite different results are obtained for n ≥ 3.

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