Page - 139 - in Joint Austrian Computer Vision and Robotics Workshop 2020

a a1 b1a2 b2 b Figure2. applying the RisingSun lemma y. UsingtheRisingSunlemma[5]onthecontinuous functionf ◦γ0 we get the shadowS= ⋃ i∈I(ai,bi) consistingofatmost countablymany intervals. S consists of the points which contradict the monotonicityoff◦γ0, thuswewant to ironout these points. Letcn :=f(an) =f(bn). Since the levelsetofcn ispath-connected,wecanconnectγ0(an)andγ0(bn) witha levelpathγ∗n : [an,bn]→A. Finally,wedefine: γ(σ) := { γ∗n(σ) σ∈ [an,bn] γ0(σ) elsewhere γ isamonotonicpathfromx toy, thusA isaslope region. 3. Results In The Plane And Counterexam- ples In HigherDimensions There are two theorems ([3] Lemma 1 and [3] Lemma 2) that are useful, but only hold if Ω⊂R2, not in general if Ω⊂ Rd for d > 2. But first, we prove a lemma. Theorem 3.1. LetA be a slope region. Then the closure A¯ isalsoasloperegion. Proof: Follows fromcontinuityoff. The following theorem is only formulated for closed slope regions, but because of the above the- oremthis isnot abig restriction. Theorem 3.2. Let d= 2 andA⊂Rd be a closed and bounded slope region. Let (∂Ai)i∈I be an enu- meration of the connected components of the bound- Figure3. a sketchof the situation inTheorem 3.2 ary∂A. For i∈ I, if∂Ai is homeomorphic to a cir- cle, then f|∂Ai has at most one local minimum and one localmaximum(but theextremamightbespread out inaconnectedplateau). Proof: Assume there are two local minima a1,a2 ∈ ∂Ai with f(a1) ≤ f(a2). Since ∂Ai is homeomorphic to a circle, there have to be lo- cal maxima b1 and b2 ∈ ∂Ai between them with f(a2)<f(b1)≤f(b2), oneoneacharc. SinceA is a slope region,a1 anda2, as well as b1 and b2 have to be connected by a monotonic path. Because of the Jordan Curve Theorem [2, p.169], these paths have to cross in a point c∈A. But this yields a contradiction: f(c) ≤ f(a2) < f(b1) ≤ f(c). Thus the assumption of the existence of two localminimahas tobe false. Note: The circle assumption is actually unnecce- sary and the proof without it remains the same in spirit, but becomes inhibitivly technical, which is whyweomit it here. Example 3.3. Let Ω =R3 andA=B1(0,0,0) be the closed unit ball. Let f be the distance to thex- Axis. f :R3→R : (x,y,z) 7→ √ y2+z2 The levelsets off inAare either thex-Axis forf≡ 0 or the sides of cylinders for f > 0. In any case, they are connected. Thus, by Theorem 2.11,A is a slope region. ∂A has one connected component, which is theunit sphere. f|∂Ahas two localminima, which are the intersections with thex-Axis, (1,0,0) and (−1,0,0). Thus, the previous theorem does not hold inR3. In fact, it does not hold in anyRd ford> 2. There isalsono limiton thenumberof localminimaonthe surfaceofa slope region. Theorem 3.4. Let d = 2 andA ⊂ Rd be a slope region. Lets∈Abeasaddlepoint. Then,s∈∂A. 139