Bounding regions to plane steepest descent curves of quasi convex families

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 the boundary 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.


Introduction
Let u be a smooth function defined in a convex body Ω ⊂ R n . Let Du(x) = 0 in {x ∈ Ω : u(x) > min u}. A classical steepest descent curve of u is a rectifiable curve s → x(s) solution to dx ds = Du |Du| (x(s)).
Classical steepest descent curves are the integral curves of a unit field normal to the sublevel sets of the given function u. We are interested in steepest descent curves that are integral curves to a unit field normal to the family {Ω t } := {x : u(x) ≤ t} of the sublevel sets for a quasi convex function u (see Definition 2.4); {Ω t } will be called a quasi convex family as in [6]. Sharp bounds about the length of the steepest descent curves for a quasi convex family, have been proved in [8], [11], [12]. The geometry of these curves, equivalent definitions, related questions and generalizations have been studied in [1], [3], [4], [9], [10].
In the above works, it has beeen proved that steepest descent curves for a quasi convex family (SDC) can be characterized as bounded oriented rectifiable curves γ ⊂ R n , with a locally lipschitz continuous parameterization T ∋ t → x(t) satisfying ẋ(t), x(τ ) − x(t) ≤ 0, a.e. t ∈ T, ∀τ ≤ t; (1) ·, · is the scalar product in R n . Let an ordering be chosen on γ, according to the orientation; let us denote γ x = {y ∈ γ : y x}.
The curves γ satisfying (1) are SDC for the related quasi convex family Ω t := co(γ x(t) ), where co(A) denotes the convex hull of the set A.
The SDC could also be chacterized in an equivalent way as self-distancing curves, namely oriented ( ) continuous curves with the property that the distance of x to an arbitrarily fixed previous point x 1 is not decreasing: In [9] self-distancing curves are called self-expanding curves. With the opposite orientation these curves have been also introduced, studied and called self-approaching curves (see [8]), or self-contracting curves (see [4]). An important property that will be used later is the property of distancing from a set A: Definition 1.1. Given a set A, an absolutely continuous curve γ, T ∋ t → x(t) has the distancing from A property if it satisfies a.e. t ∈ T, and ∀y ∈ A.
Steepest descent curves (or self-distancing curves) γ that also satisfy the above property with respect to a convex set K will be called SDC K . Of course if γ is a SDC and x ∈ γ then γ \ γ x is a SDC co (γx) .
In the present work we are interested on the behaviour and properties of a plane SDC γ beyond its final point x 0 . The principal goal of the paper is to show that conditions (1) or (3) imply constraints for possible extensions of the curve γ beyond x 0 ; these constraints are written as bounding regions for the possible extensions of γ x 0 .
Let us outline the content of our work. In §2 introductory definitions are given and covering maps for the boundary of a plane convex set, needed for later use, are introduced. In §3 the involutes of the boundary of a plane convex body are introduced and some of their properties are proved.
In §4 plane regions depending on the convex body co(γ x 0 ) have been defined; these regions fence in or fence out the possible extensions of γ x 0 . The boundary of these sets consists of arcs of involutes of convex bodies, constructed in §3. As an application, in §4.1 the following problem has been studied: given a convex set K, x 0 ∈ ∂K, x 1 ∈ K is it possible to construct a SDC K joining x 0 to x 1 ? Minimal properties of this connection have been introduced and studied. In §5 sets of points more general than SDC are studied . A set σ ⊂ R 2 (not necessarily a curve) of ordered points satisfying (3) will be called self-distancing set, see also Definition 2.1; with the opposite order, σ was called self-contracting in [3] and many properties of these sets, as only subsets of self contracting curves, were there obtained. A natural question arises: does it exist a steepest descent curve γ ⊃ σ? Examples, necessary and/or sufficient conditions are given when σ consists of a finite or countable number of points x i ∈ R 2 and/or steepest descent curves γ i ⊂ R 2 .
In the present work the two dimensional case is studied. Similar results for the n dimensional case are an open problem stated at the end of the work.
A not empty, compact convex set K of R n will be called a convex body. From now on, K will always be a convex body not reduced to a point. Int(K) and ∂K denote the interior of K and the boundary of K, |∂K| denotes its length, cl(K) is the closure of K, Aff (K) will be the smallest affine space containing K; relint K and ∂ rel K are the corresponding subsets in the topology of Aff (K). For every set S ⊂ R n , co(S) is the convex hull of S.
Let q ∈ K; the normal cone at q to K is the closed convex cone When q ∈ Int(K), then N K (q) reduces to zero. The tangent cone, or support cone, of K at a point q ∈ ∂K is given by In two dimensions cones will be called sectors.
Let K be a convex body and p be a point. A simple cap body K p is: Cap bodies properties can be found in [2], [14].

Self-distancing sets and steepest descent curves
Let us recall the following definitions Definition 2.1. Let us call self-distancing set a bounded subset σ of R n , linearly ordered (by ), with the property: The self-distancing sets has been introduced in [3] with the opposite order. If a self-distancing set σ is a closed connected set, not reduced to a point, then it can be proved that σ is a steepest descent curve (see [3,Theorem 3.3], [9,Theorem 4.10]) and it will also be called a self-distancing curve γ.
The short name SDC will be used for self-distancing curves (steepest descent curves) in all the paper. Definition 2.2. Let K be a convex body, γ ⊂ R 2 \ relint K will be called a self-distancing curve from K (denoted SDC K ) if: iii) γ has the distancing from K property: When (ii) does not hold, that is γ ∩ ∂ rel K = ∅, γ will be called a deleted self-distancing curve from K. Remark 2.3. Let γ be a SDC K , then γ has an absolutely continuous parameterization, thus property (8) for γ is equivalent to (4).
Nested families of convex sets have been introduced and studied by De Finetti [5] and Fenchel [6]. Let us recall some definitions.
Definition 2.4. Let T be a real interval. A convex stratification (see [5]) is a not empty family K of convex bodies Ω t ⊂ R n , t ∈ T ⊂ R, linearly strictly ordered by inclusion (Ω 1 ⊂ Ω 2 , Ω 1 = Ω 2 ), with a maximum set (max K) and a minimum set (min K).
Let K = {Ω t } t∈T be a convex stratification. If for every s ∈ T \ {max T } the property: t>s Ω t = Ω s holds, then as in [6], K = {Ω t } t∈T will be called a quasi convex family.
An important quasi convex family associated to a continuous self-distancing curve from K, γ: The couple (γ, K) is special case of Expanding Couple, a class introduced in [9].

The support function of a plane convex body
Let K ⊂ R n be a convex body not reduced to a point. For a convex body K, the support function is defined as where ·, · denotes the scalar product in R n . For n = 2, ϑ ∈ R, let θ = (cos ϑ, sin ϑ) ∈ S 1 and h K (ϑ) := H K (θ), it will be denoted h(ϑ) if no ambiguity arises. For every θ ∈ S 1 there exists at least one point x ∈ ∂K such that: this means that the line through x orthogonal to θ supports K. For every x ∈ ∂K let N x the set of θ ∈ S 1 such that (10) holds. Let F (θ) be the set of all x ∈ ∂K satisfying (10). If ∂K is strictly convex at the direction θ then F (θ) reduces to one point and it will be denoted by x(θ).
Definition 2.7. The set valued map: G : ∂K → S 1 , ∂K ∋ x → N x ⊂ S 1 , is the generalized Gauss map; x ∈ ∂K is a vertex on ∂K iff N x is a sector with interior points. The set valued map F : is the reverse generalized Gauss map; F (θ) is a closed segment, possibly reduced to a single point, and it will be called 1-face when it has interior points.
Let P be the covering map be the parametric representations of ∂K depending on the arc length counterclockwise (clockwise) with an initial point (not necessarily the same). Let us extend x l (·) and x r (·) by defining Similarly for x r .
Let us fix For later use, we need to have x 0 = x l (s 0 ) = x r (s 0 ); this can be realized by choosing suitable initial points for the parameterizations x l and x r . Then The maps x l : R → ∂K, x r : R → ∂K, are covering maps.
The function s l+ is increasing in R, right continuous and with left limits (so called cadlag function). Similar properties hold for −s r− . Let us recall that a cadlag increasing function s(ϑ), ϑ ∈ R has a right continuous inverse defined as ϑ(s) = inf{ϑ : s(ϑ) > s}.
Let ϑ → h(ϑ) be the support function of K.
It is well known ( [7]) that, if ∂K is C 2 + (that is ∂K ∈ C 2 , with positive curvature), then h is C 2 and the counterclockwise element arc ds of ∂K is given by ds = (h +ḧ)dϑ.
h(ϑ) +ḧ(ϑ) is the positive radius of curvature; moreover the reverse Gauss map F : θ → x ∈ ∂K is a 1-1 map given by The previous formula also holds for an arbitrary convex body, for every ϑ such that F (θ) is reduced to a point, see [2]. Let us recall that a real valued function x → f (x) is called semi convex on R when there exists a positive constant C such that f (x) + Cx 2 is convex on R. From (15) the function ϑ → h(ϑ) + 1 2 ϑ 2 max h is convex on R, thus h is semi convex. In the case that K is an arbitrary convex body, by approximation arguments with C 2 + convex bodies, see [14], it follows that the support function of K is also semi convex. As consequence h is Lipschitz continuous, it has left (right) derivativeḣ − (respectivelyḣ + ) at each point, which is left (right) continuous. Moreover at each point the right limit ofḣ − isḣ + and the left limit ofḣ + isḣ − , see [13, pp. 228].
It is not difficult to show (from (16), with a right limit argument) that for an arbitrary convex body, for ϑ ∈ R, the formula holds. Similarly the formula holds.
If ∂K is not strictly convex at the direction θ = (cos ϑ, sin ϑ) then h is not differentiable at ϑ anḋ If x 1 , x 2 ∈ ∂K let us define arc + (x 1 , x 2 ) the set of points of ∂K between x 1 and x 2 according to the counterclockwise orientation of ∂K, and arc − (x 1 , x 2 ) the set of points between x 1 and x 2 , according to the clockwise orientation; |arc ± (x 1 , x 2 )| denote their length.
Remark 2.8. It is well known that a sequence of convex body K (n) converges to K if and only if the corresponding sequence of support functions converges in the uniform norm, see [14, pp. 66]. Moreover as the two sequences of the end points of a closed connected arc of ∂K (n) converge, then the sequence of the corresponding arcs converges to a connected arc of ∂K and the sequence of the corresponding lengths converges too.
Proposition 2.9. Let K be a convex body and h its support function, then Proof. For every convex body K not reduced to a point the function ϑ → s l+ (ϑ) is defined everywhere and satisfies the weak form of (15), namely: Using the fact that ϑ → h(ϑ) is Lipschitz continuous, integrating by parts (22), the formula with c constant. Passing to the right limit, the equality holds. The formula (20) follows, by computing s l+ (ϑ) − s l+ (ϑ 0 ), using the previous equality. Similarly (21) is proved.
3 Involutes of a closed convex curve Let us notice that s is the arc length of the curve, not of the involute; if s 0 = 0, then the starting point of the involute coincides with the starting point of the curve. It is easy to construct an involute of a convex polygonal line (even if the classical definition (24) does not work) by using arcs of circle centered at its corner points; moreover the involute depends on the orientation of the curve.
In this section, involutes for the boundary of an arbitrary plane convex body K, not reduced to a point, will be defined. The assumption that K is an arbitrary convex body is needed to work with the involutes of the convex sets, not smooth, studied in §4.
Let K ∈ C 2 + ; let x 0 be a fixed point of ∂K, s → x(s) can be the clockwise parameterization of ∂K or the counterclockwise parameterization. Since there exist two orientations, then two different involutes have to be considered. As noted previously one can assume that the parameterizations of ∂K have been chosen so that x 0 = x l (s 0 ) = x r (s 0 ). Definition 3.2. Let us denote by i l,x 0 the left involute of ∂K starting at x 0 corresponding to the counterclockwise parameterization of ∂K, by i r,x 0 the right involute corresponding to the clockwise parameterization. When one needs to emphasize the dependence on K of involutes, they will be written Let us notice that if ρ is a plane reflection with respect to a fixed axis then ). This relation allows us to prove our results for the left involutes only and to state without proof the analogous results for the right involutes.
Theorem 3.4. Let us fix the initial parameters x 0 , s 0 , θ 0 , ϑ 0 . The left and the right involutes of a plane convex curve starting at x 0 ∈ ∂K, boundary of a C 2 + plane convex body K with support function h, are parameterized by the value ϑ related to the outer normal n ϑ to K, as follows Proof. In the present case there is a 1-1 mapping between ϑ and s; from (15), it follows then, changing the variable s with ϑ in (24), with elementary computation, (25) is obtained (since x ′ (s) = t ϑ and (16) holds). Formula (26) follows from (18) and (21).
For an arbitrary convex body K in place of (16), formulas (17),(18) have to be used. Definition 3.5. Let K be a plane convex body, let The left involute of ∂K starting at x 0 will be defined as , the right involute starting at x 0 will be defined as Similarly from (28), (21) it follows that (30) Let us notice that in (29), (30) the same parameter ϑ is used, but with different range; it turns out that i l is counterclockwise oriented; instead i r is clockwise oriented; Remark 3.6. The following facts can be derived from the above equations: i) since h is Lipschitz continuous for every convex body K, then the involute i l,x 0 is a rectifiable curve; Lemma 3.7. The parameterization (29) of the involute i l,x 0 is 1-1 in the interval [ϑ + 0 , ϑ + 0 + 2π); moreover, except for at most a finite or countable set F of values ϑ i , i = 1, 2, . . . ... (corresponding to the 1-faces F θ i of ∂K), i l,x 0 is differentiable and: furthermore i l,x 0 has left and right derivative with common direction n ϑ at ϑ = ϑ i ∈ F.
Proof. By differentiating (29) and using (20), the equality (33) is proved. Similar argument, at ϑ = ϑ i ∈ F, proves that n ϑ is the common direction of the left and right derivatives.
then they will be called parallel curves. Moreover, by (32), i l,x 0 (ϑ) and i l,x 0 (ϑ + 2π) will also be called parallel.
the involute is a convex curve with positive curvature a.e.
Moreover the following properties hold.
i) For every σ > 0 the right derivative exists everywhere and it is a decreasing cadlag function; ii) d dσ i l,x 0 has everywhere right derivative given by Theorem 3.10. Let K (n) be a sequence of plane convex bodies which converges uniformly to K, x (n) ∈ ∂K (n) , x (n) → x 0 ; then the corresponding sequences of left involutes i K (n) l,x (n) converge uniformly to i l,x 0 in compact subsets of [ϑ + 0 , +∞]; moreover the corresponding sequence of their derivatives (with respect to the arc length) converges uniformly to d dσ i l,x 0 .
Proof. By Remark 2.8 the sequence of functions s n l+ converge to s l+ . From (34) the arclengths of the left involutes i K (n) and the set valued map F (Definition 2.7). Let the union of segments joining the points of η with the corresponding points on ∂K.
Definition 3.11. If the tangent sector T (x 0 ) to K has an opening less or equal than π/2 as in Fig.1, then Q ∪ K is convex; let us define Theorem 3.12. Let i l := i l,x 0 be the left involute starting at x 0 on the boundary of a plane convex body K, then: iv) if y ∈ ∂K, then J y (ϑ) is not decreasing for ϑ ≥ ϑ + 0 and d dϑ J > 0 for (cos ϑ, sin ϑ) ∈ N K (y).
Proof. As i l is rectifiable, then the function J 2 y (ϑ) = |i l (ϑ) − y| 2 is an absolutely continuous function for ϑ ≥ ϑ + 0 , and from (33) for ϑ ∈ F the last inequality holds since n ϑ is the outer normal to ∂K at x l (s l+ (ϑ)). Moreover the previous inequality is strict for all ϑ if y ∈ Int(K), it is also a strict inequality for y ∈ ∂K and y ∈ F (θ). This proves iii) and iv). Then i) follows from iii) and Definition 1.1 of distancing from K property for a curve. To prove ii) let us recall that a SDC satisfies (1); then one has to prove that the angle at i l (ϑ) between the vector i l (ϑ) − i l (τ ), ϑ + 0 < τ < ϑ ≤ ϑ * l , and n ϑ , the tangent vector at i l (ϑ), is greater or equal than π/2; this is equivalent to show that the half line r ϑ through i l (ϑ) and x(s l+ (ϑ)) orthogonal to n ϑ supports at i l (ϑ) the arc of i l from x 0 to i l (ϑ). By Definition 3.11 this is the case for all ϑ between ϑ + 0 and ϑ * l .
Proof. From i) of Theorem 3.12 the left involute is a curve such that the distance of its points from all y ∈ K is not decreasing; ii) of the same theorem proves that it is a SDC. Let us recall that a self-contracting curve is a self-distancing curve with opposite orientation.
Theorem 3.14. Let K be a plane convex body not reduced to a single point and let x 0 , s o , θ 0 , ϑ 0 be the initial parameters. Let [ϑ + 0 , ϑ + 0 + 2π] ∋ ϑ → i l (ϑ) be an arc of the left involute starting at be an arc of the right involute ending at x 0 ; then there exists only one point y = x 0 which belongs to both arcs and with Proof. For simplicity, first let us prove the existence of y assuming that K ∈ C 2 + . With the assumed conditions , R ∋ ϑ → x(ϑ) := x(θ) defined by (16) is a parameterization of ∂K.
Thus P + (ϑ l ) is on both arcs of involutes and the other way around.
As i l (ψ(ϑ)) − x(ϑ) = λt ϑ (λ > 0) and (40) holds, then ψ ′ > 0, ψ is strictly increasing and continuously differentiable. Let us prove (ii). The formula holds. Let us notice that i l (ϑ) − i l (ψ(ϑ) is parallel to t ϑ ; thus by (33) On the other hand As the angle between t ϑ and n ψ(ϑ) is acute, then last term in the above equalities is positive; thus the derivative in the left hand side of (42) is positive and (ii) of Claim 2 follows. Claim 3: In the interval [ϑ 0 , ϑ * l ] the function φ has values smaller than L and greater than L.
Let ρ be the half line with origin x 0 and direction −t ϑ 0 ; ρ − {x 0 } crosses the arc i r in a first point y 1 = i r (α 1 ), with α 1 < ϑ 0 − π/2. Then The half line ρ meets the arc i l in a point y 2 and |y 2 − x 0 | = L.
Let us prove now that the point y is unique. Let us argue by contradiction. Let P, Q be two distinct points on i l ∩ i r , with P ≺ Q on i l and i r ; then since i l is a distancing curve from x 0 : and since i r is a contracting curve to x 0 : Therefore all the points on the arc of i l and of i r between P and Q have the same distance from x 0 ; thus, between P and Q, i l and i r (arc of involutes of a same convex body K) coincide with the same arc of circle centered at x 0 , this implies that K reduce to the point x 0 , which is not possible for the assumption.
Definition 3.15. Let z ∈ K. Let z l (z r ) ∈ ∂K on the contact set on the "left" (right) support line to K through z. If the contact set is a 1-face on these support lines, then z l and z r are identified as the closest ones to z. The triangle zz l z r is counterclockwise oriented. Theorem 3.16. For every ξ ∈ ∂K let us consider the left involutes i l,ξ and the right involutes i r,ξ parameterized by their arc length σ. The maps Proof. Assume, in the proof, that x 0 ∈ ∂K, θ 0 ∈ G(x 0 ), ϑ 0 , s 0 are fixed. Let z ∈ K. The tangent sector to the cap body K z with vertex z has two maximal segments zz l , zz r on the sides that do not meet K (except at the end points z l ,z r ). Let ϑ l such that z l = x l (s l+ (ϑ l )), and let s such that |z − z l | = s l+ (ϑ l ) − s.
Let ξ l = x l (s), let ϑ = ϑ + l (s). From (31) and from the definition of left involute (27) (with ξ l in place of x 0 , ϑ in place of ϑ + 0 , s in place s 0 ) holds; thus the map (ξ, σ) → i l,ξ (σ) is surjective. Moreover the map it is also injective, since the left involutes don't cross each other since they are parallel (see Remark 3.8). Similar proof holds fir the right involutes.
Let ξ l = x l (s) be the starting point of the left involute i l,ξ l through z, defined in the previous theorem; similarly let ξ r be the starting point of the right involute i r,ξr through z. Let us notice that i l,ξ l and i r,ξr meet each other in a countable ordered set of points.

J-fence and G-fence
be the first point where the two involutes cross each other (see Theorem 3.14). Let us define J(K, x 0 ) will be called the J-fence of K at x 0 .
Let us notice that J l (K, x 0 ) and J r (K, x 0 ) are two convex bodies with in common the segment x 0 y only.
From Theorem 3.16 the starting point ξ l (ξ r ) of a left(right) involute is uniquely determined from any point z ∈ K of the involute. The arc of the points on the left (right) involute between the starting point and z will be denoted by i z l,ξ l (i z r,ξr ), or i z l (i z r ) for short. For y w let us denote with i y,w l (i y,w r ) the oriented arc of the left (right) involute between y and w. Let us introduce now other regions which are bounded by left and right involutes.
Definition 3.18. Given z ∈ R 2 \ K, let i l = i l,ξ l (i r = i r,ξr ) be the left (right) involute through z with starting point ξ l (ξ r ) and let z l (z r ) ∈ ∂K be as in Definition 3.15. Let ϑ + ξ l satisfying x l (s l+ (ϑ + ξ l )) = ξ l . Let ϑ l > ϑ + ξ l be the smallest angle for which x l (s l+ (ϑ l )) = z l . Let us consider the parameterization (27); let us define is an open, bounded, connected set. G(K, z) will be called the G-fence of K at z.

Remark 3.19.
If z is the first crossing point of i l and i r and ξ l = ξ r , then G(K, z) = Int(J(K, ξ l )).
Let us conclude this section with the following result, which follows from Theorem 3.10.

Bounding regions for SDC in the plane
Let us assume that x 0 is the end point of one of the following sets a) a steepest descent curve γ, satisfying (1) and (3); b) γ K : a self-distancing curve from a convex body K, see Definition 2.2.
The following questions arise: can one extend γ, γ K beyond x 0 ? Which regions delimit that extension? Which regions are allowed and which are forbidden?
Lemma 4.1. Let z ∈ R 2 \ K. If u ∈ G l (K, z) then the arc i u l of the left involute to K ending at u, is contained in G l (K, z). Similarly if u ∈ G r (K, z), then i u r ⊂ G r (K, z).
Then the arc i u l is parallel to an arc of the left involute i l (through z) for ϑ ∈ (ϑ + ξ l , ϑ l ). Then any left tangent segment to K from a point of i u l is contained in the left tangent segment from the corresponding point of i z l .
Lemma 4.2. Let z ∈ R 2 \ K and let u ∈ G l (K, z). There are two possible cases: i) if the right involute ending at u does not cross the tangent segment z l z or it crosses z l z at a point q ∈ G l (K, z), then in both cases i u r ⊂ G l (K, z); ii) if the right involute ending at u crosses the tangent segment z l z at a point q ∈ z l z ∩ ∂G l (K, z), then i q,u r \ {q} ⊂ G l (K, z).
Proof. Since the starting point ξ r (u) of the right involute ending at u is on ∂K, the distance from ξ r (u) to a point of the left involute i z l is not decreasing, see iv) of Theorem 3.12; similarly the distance from ξ r (u) to a point of i u r is not decreasing. In the case i) the arc i u r has its end points in G l (K, z) and by the above distance property it can not cross two times the left involute, then it can not cross the boundary of G l (K, z), therefore i u r ⊂ G l (K, z); similarly in the case ii) the arc i q,u r can not cross the boundary of G l (K, z) at most than in q; therefore all the points of this arc, except than q, belong to G l (K, z).

From the previous lemma it follows that
Proof. By Lemma 4.1 the left involute that bounds G l (K, u) is inside G l (K, z)), then (46) is proved. Inclusion (47) is proved similarly. Let u ∈ G(K, z) = Int(cl(G l (K, z) ∪ G r (K, z))) and let us consider u ∈ G l (K, z), then in case i) of Lemma 4.2 also the open arc of the right involute i u r is inside G l (K, z)) ⊂ G (K, z)). Besides i u l ⊂ ∂G l (K, u), then (48) is trivial. In case ii) of Lemma 4.2 the open arc i q,u r is inside G l (K, z)). On the other hand q is inside G r (K, z)) and by (47) the arc i q r ⊂ i u r is in G r (K, z)) ⊂ G (K, z)). Similar arguments hold if u ∈ G r (K, z)). Then (48) holds in this case too.
Proof. To prove (49), let us assume, by contradiction, that η has a point z ∈ G(K, w). With no loss of generality it can be assumed that z ∈ ∂G(K, w) and η \ η z ⊂ G(K, w).
Then, z is the end point of a segment zw i , where w i ∈ G(K, w) ∩ η and z ≺ w i on η. As z ∈ ∂G(K, w), then there exists an involute through z which is a piece of the boundary of G(K, w) (to fix the ideas it is assumes that it is the left involute i l ). Let us consider z l ∈ ∂K so that the tangent vector t z to i l at z satisfies t z , z − z l = 0.
As w i is inside the orthogonal angle centered in z with sides t z and z l − z, then w i − z, z − z l < 0.
Then as for ε > 0 sufficiently small, z ε := z + ε(w i − z) ∈ η w i and at z ε the curve η has tangent vector w i − z that satisfies contradicting the fact that η w has the distancing from K property (4). This proves (49). If w n → y, with y ∈ G(K, w n ), also the inclusions hold. Then (50) is obtained by the approximation Theorem 3.20.
Theorem 4.5. Let K be a convex body and let γ K be SDC K , w ∈ γ, w ∈ K. Then γ K w ⊂ cl(G(K, w)).
Proof. Let us choose a sequence {w n }, w n ∈ γ K , w n w, w n → w. Let us fix the arc γ K wn . By [9, Theorem 6.16], γ K wn is limit of SDC K polygonals with end point w n . From Lemma 4.4, these polygonals are enclosed in cl(G(K, w n )); then γ K wn ⊂ cl(G(K, w n )) holds too. The inclusion (51) is now obtained from the limit of the previous inclusions and by the approximation Theorem 3.20.
Theorem 4.6. Let K be a convex body not reduced to a point. If γ K is a self-distancing curve from K with starting point x 0 ∈ ∂K, then Proof. Let z be the first crossing point of the left and right involutes of K starting at x 0 . Then Int(J(K, x 0 )) = G(K, z).
By contradiction, if γ K has a point w ∈ G(K, z), then, by Theorem 4.5, the following inclusion holds since, by the distancing from K property, γ K has in common with K only the starting point x 0 then, the following inclusion γ K w \ {x 0 } ⊂ cl(G(K, w)) \ ∂K holds too. Moreover by (48) the set cl(G(K, w)) \ ∂K has positive distance from the R 2 \ G(K, z); then γ K w \ {x 0 } has a positive distance from R 2 \ G(K, z) = R 2 \Int(J (K, x 0 )). This is in contradiction with x 0 ∈ ∂J(K, x 0 ). Corollary 4.7. Let γ be a SDC and z 1 ∈ γ then γ \ γ z 1 ⊂ cl(R 2 \ J(co(γ z 1 ), z 1 )).
Proof. Since γ \ γ z 1 is a self-distancing curve from co(γ z 1 ) and z 1 ∈ ∂co(γ z 1 ) (see [9, (i) of Lemma 4.6], then Theorem 4.6 applies to γ K = γ \ γ z 1 with K = co(γ z 1 ). Definition 4.8. Let γ be a SCD. If z 1 , z ∈ γ, with z 1 z let For z ∈ K, let K z be the cap body, introduced in (6). Next theorem shows the principal result on bounding regions for arcs of a SDC γ. Theorem 4.9. Let K be a convex body and let γ be a SDC K . If z 1 , z ∈ γ, with z 1 z then γ z 1 ,z ⊂ cl(G(K, z) \ J(K z 1 , z 1 )).
Proof. First let us notice that γ z 1 ,z has the distancing from K and from the set point {z 1 } property, thus by Proposition 2.6 it has the distancing from K z 1 property. Then the inclusion (53) follows from Theorems 4.5 and 4.6.
Let us conclude the section with the following inclusion result for Jfences.
Theorem 4.10. Let K, H be two convex bodies not reduced to a point, Proof. The boundary of J(H, x 0 ) consists of two arcs of the left and right involutes of H starting at x 0 . By Corollary 3.13 they are SDC H , then they are SDC K ; therefore by Theorem 4.6 they cannot intersect the boundary of J(K, x 0 ). Given a point x 1 ∈ K, the segment joining it with its projection x 0 on ∂K is a SDC K which minimally connects the two points.

Miniminally connecting plane steepest descent curves
This subsection is devoted to consider when it would be possible to connect a given point x 0 on the boundary of a plane convex body K, with an arbitrarily given point x 1 ∈ K, by using a steepest descent curve γ ∈ SDC K . Let us denote with Γ K x 0 ,x 1 the class of the curves γ ∈ SDC K starting at x 0 and ending at x 1 .
Definition 4.11. Let γ be a SDC with end point y and η be a SDC with starting point y; let us denote by γ ⋆ η, the curve joining γ with η in the natural order, if it is a SDC curve.
Let us prove now that (54) is sufficient. Let us notice that R 2 \(J(K, x 0 )∪ K) can be divided in four regions N, B l , B r , V , see Fig. 4, defined as follows i) the closed normal sector N := x 0 +N K (x 0 ) is the angle bounded by the two half lines t l , t r tangent at x 0 ∈ ∂K to the left and right involute i l := i l,x 0 , i r := i r,x 0 respectively; this angle can be reduced to an half line, starting at x 0 ; ii) let P be the first crossing point between i l and i r , see Theorem 3.14; i l is a SDC until to i l (ϑ * l ), which will be a point Q l following P ; after Q l the involute i l is no more a SDC, see i) of Theorem 3.12.
Let us change i l after Q l with j l,Q l , the left involute of co(K ∪ i Q l l ) at Q l . Let us define P l be the first intersection point of j l,Q l with ∂N , and letĩ P l l,x 0 := i Q l l,x 0 ⋆ j P l l,Q l .
It is not diffcult to see thatĩ P l l,x 0 ∈ Γ K x 0 ,P l . Changing the left with the right,ĩ r,x 0 and the point P r can be constructed. Let B r the union of the arc i P r,x 0 \ {P } with the plane open region bounded by the segment x 0 P l , the arc i P r,x 0 and the arcĩ P,P l l ; let B l the union of the arc i P l,x 0 \{P } with the plane open region bounded by the segment x 0 P r , the arc i P l,x 0 and the arcĩ P,Pr On the oriented curveĩ Pr r,x 0 there are two points so that their tangent lines contain x 1 . Let y * r be the first tangency point. Let y l , y r be the intersection points of the half line m starting at x 0 and containing x 1 , withĩ P l l,x 0 and withĩ Pr r,x 0 respectively, see Fig. 4. Under the assumption (54), x 1 belongs to one of the four regions N, B l , B r , V ; let us prove now (55), (56) in the four corresponding cases.
The curve η r is a SDC K , since the normal lines at all the points on the segment y * r x 1 have the same directions and supportĩ y * r r,x 0 up to y * r ; then η r is a SDC K and joins Thus Γ K x 0 ,x 1 is not empty and contains at least the two elements η l , η r . Let us consider the connected closed curve c x 1 :=ĩ yr r,x 0 ∪ y r y l ∪ĩ y l l,x 0 .
Let γ ∈ Γ K x 0 ,x 1 , let T ∋ t → x(t) ∈ γ be a continuous parameterization of γ. Let us project from x 0 the curve γ on c x 1 and let D be this projection. That is, for t ∈ T , let λ t := {x 0 + λx(t), 0 ≤ λ} and let Clearly D is a closed connected subset of c x 1 containing x 0 and the segment y r y l . Thus D contains at least one of the two connected components of c x 1 joining x 0 with the y r , y l . Therefore the inclusions (ĩ yr r,x 0 ∪ y r y l ) ⊂ D, (or both) hold.
Assume that (60) holds and let η r be defined as in (57). Since, by construction of D, the set co(D ∪ {x 1 }) is contained in co(γ), then Similarly if (61) holds, then 3. Let x 1 ∈ B r ; the same argument as in the case 2. can be carried on up to the inclusions (60), (61). As in the step 2., when case (60) holds, the curve η r can be constructed and η r is a SDC K .
Let us show that if x 1 ∈ B r then (61) cannot occur, so the curve η l can not to be constructed.
Moreover γ \ γ z 1 ∪ {z 1 } is a SDC co(K∪γz 1 ) , see Remark 2.5. Then by (62) it is a SDC co(K∪J l (K,x 0 )) . Let us consider the convex body H = co(K ∪ i P l,x 0 ). Since Let P 1 where the right tangent from z 1 to H crosses the arcĩ P l l,P . Let us notice thatĩ P l l,P is also an arc of the left involute of H at P . Moreover i P 1 ,P l l is an arc of the left involute of H z 1 starting at P 1 , then it is parallel to the left involute of H z 1 at z 1 until it crosses the sector N ; it turns out that B r ⊂ J(H z 1 , z 1 )).
From the two previous inclusions a contradiction comes out since Then (61) cannot occur.
4. The case x 1 ∈ B l is similar to the previous one.
The proof is complete.
Definition 4.13. Under the assumptions of Theorem 4.12, let us define E K x 0 ,x 1 the set of η i , i = 1, 2 (possibly coinciding) as they are constructed in the proof of Theorem 4.12, which satisfy (55),(56). These curves will be called minimally connecting steepest descent curves for the class Γ K x 0 ,x 1 .
Definition 4.14. Let γ : T ∋ t → x(t) be an absolutely continuous curve and let x(t) be a point of γ, with tangent vectorẋ(t). Let is an half plane bounded by the normal line to γ at x(t) and it is defined almost everywhere in T . For the curve γ \ γ x 1 (consisting of the points of γ following x 1 ) let us define the region: If H(γ, x 1 ) = ∅, then it is a convex set.
5 Self-distancing sets and steepest descent curves .
A self-distancing set σ will be called SDC-extendible if there exists a steepest descent curve γ such that σ ⊂ γ.
This section is devoted to investigate the following question: Can a self-distancing set σ be extended to a steepest descent curve γ? Let us call Γ σ the family of SDC γ wich extends σ. The following example shows that Γ σ can be empty.
Example 5.1. Let us consider in a coordinate system xy the points: .
Proof. By contradiction let γ ∈ Γσ, then any point x on the arc γ x 3 ,x 4 satisfies the inequalities That is x ∈ ∂B(x 1 , 3) and consists of two points only, that is impossible.
Next theorem gives a necessary condition (64) in order to extend a finite self-distancing set σ to a SDC; this condition is based on the bounding sets introduced in §3.1.
Let us define σ x as the subset of σ consisting of the point x and of the previous ones on σ (consistently with (2)).
Theorem 5.2. Let σ be a self expanding SDC-extendible set, then for all x 0 ∈ σ such that σ x 0 = {x 0 }, the following inclusion holds.
Remark 5.3. In the Example 5.1 it has been proved, in a simple way, thatσ is not SDC-extendible. Another way to prove this fact is to check that the condition (64) does not hold for the point x 4 ; let us notice that consists of a circular arc centered at x 1 with radius 2 + 13 − 4 √ 8; then it is easy to see that x 4 is in the interior of J(co(σ x 3 ), x 3 ) and (64) is not satisfied.
Let us show in the following example that (64) is not sufficient for a self-distancing set σ to be SDC-extendible.
This is in contradiction with Corollary 4.7 at the point ξ 3 .
Let us introduce definitions and preliminary facts needed to obtain necessary and sufficient conditions for the extendibility of a self-distancing set σ structured as follows.
Definition 5.5. Let us denote with∪ i σ i a self-distancing set with a finite (or countable) family of closed connected components σ 1 , σ 2 , . . . , σ n , . . ., ordered as the points of σ, that is if i < j, x ∈ σ i , y ∈ σ j ⇒ x y. Let x − i be the first point and let x + i be the last point of σ i ; if they are distinct (that is σ i does not reduce to a point) as noticed in the introduction ([3, Theorem 3.3] and [9, Theorem 4.10]), σ i is a SDC and it will denoted by γ i . Lemma 5.6. Let σ =∪ i σ i be a self-distancing set. A necessary condition for Γ σ = ∅ is that for all components σ i , which are curves γ i , the following fact holds.
Let us start to study a self-distancing set with two closed connected components.
Proof. case p1) is trivial; in the case p2) the inclusion (69) follows from Lemma 5.6 with i = 2 in (65). It is also trivial that it is sufficient. The case p3) follows from Theorem 4.12 with K = co(γ 1 ). The case p4) follows from Theorem 4.15 with γ 2 in place of γ 1 , K = co(γ 1 ), x + 1 in place of x 0 , x − 2 in place of x 1 .
An easy sufficient condition to check if σ =∪ i σ i is extendible is the following Theorem 5.12. Let σ =∪σ j be a self expanding set and let E 2 , E 3 , . . . , E i , . . . be the sequence (finite or countable) of the essential families associated to σ. Then Γ σ = ∅ iff ∀i ≥ 2 the essential family E i is not empty.
Open problem: In the present work only two dimensional problems are studied. In three (or more) dimensions the construction of boundary regions to a SDC and to a SDC K is open. The boundary regions should probably be constructed by using the space involutes of the geodesics curves on ∂K.