ARSTRACT. Plane, oriented, rectifiable curves, such that, in almost each x the normal line bounds a half-plane containing the part of the curve preceding x, are considered. It is shown that, in the family of curves as above, with convex hull of given perimeter, there exist curves of maximal length and these are evolutes of themselves. As a consequence, it is proved that quasi-convex functions in a set f~ have steepest descent lines with length bounded by the diameter of f~. This result is then extended to R".

Let u i and u 2 be two Solutions of the heat equation u t -u xx = 0, defined in {(*, t):Q 0, then 5j (t) is close to s 2 (t) for / > 0. 1980 Mathematics Subject Classification (1985 Revision): 35K05. 0. Introduction It is well known that the Solutions of the heat equation do not depend continuously on the noncharacteristic Cauchy data. However, suitable a-priori bounds can restore the continuous dependence (see [Be], [Kn-Vel], [Kn-Ve2], [La-Ro-Si], [Ma-Mi], [Mu], [Pa], [Pu]) of the Solutions, studied in a fixed domain.It is pretty natural to look for the continuous dependence upon Cauchy data of Solutions to the equation above, that have different domains of existence. A more precise Statement of the problem could be the following: let w l5 u 2 two Solutions of the heat equation, each defined in a region {(je, t):Q < t < T,0 0 and έ?(ί) = l/ε, s$(i) = l/2ε. The functions u ( l\x, t) = l -εχ, u 2 (x, t) = 1 -Lavoro eseguito nelFambito de o I.A.G.A. -C.N.R.

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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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