Boundaries of Smooth Strictly Convex Sets in the Euclidean Plane 2 R

We give a characterization of the boundaries of smooth strictly convex sets in the Euclidean plane 2 R based on the existence and uniqueness of inscribed triangles.


Introduction
The reader unfamiliar with the theory of convex sets is referred to the books [1] [2] [3] [4] [5].Let M be a set in the n -dimensional Euclidean space n R .In the following we shall denote by int M , clM , M ϑ , convM the interior, the closure, the boundary and respectively the convex hull of the set M .With .A convex curve is a connected subset of the boundary of a convex set.

Preliminaries
In the chapter 8 of the book [4] of F.A. Valentine the author says the following: "It is interesting to see what kind of strong conclusions can be obtained from weak suppositions about any triplet of points of a plane set S ."In [6] Menger gives such a characterization of the boundary of a convex plane set S based on intersection properties of S with the seven convex sets in which the space 2  R is subdivided by the lines ( ) ( ) , , , L x x L x x and ( ) , , , , A survey of different characterizations of convex sets is given in the paper [8].The results of K. Menger and that of K. Juul give characterizations of the boundaries of convex sets.
In the years 1978 [9] and 1979 [10] we have proved the following two theorems giving a characterization of the boundaries of smooth strictly convex sets: Theorem 2. A plane compact set S is the boundary of a smooth strictly convex set if and only if the following two conditions hold: 1) 2) For every triangle ( ) R there is one and only one triangle , , p p p ′ ′ ′ ∆ homothetic to the triangle ( )

Main Results
The main result of this paper is Theorem 4 giving another characterization of the boundaries of smooth strictly convex sets in the Euclidean plane 2 R which uses also condition (2) of the Theorem 2 and Theorem 3. Theorem 4. A compact set S in the Euclidean plane 2 R is the boundary of a smooth strictly convex set if and only if there are verified the following three conditions: 1) For every triangle ( ) R there is one and only one triangle ( ) , , p p p ′ ′ ′ ∆ homothetic to the triangle ( ) , , p p p ∆ inscribed in the set S , i.e. such that 1 2 3 , , p p p S ′ ′ ′ ∈ .
2) For any two distinct points p S ∈ and q S ∈ there are at least two points R by the line ( ) 3) The set S does not contain three collinear points.
For the proof of Theorem 4 we need the following theorem from the paper [11] and three lemmas: that S is a strictly convex closed arc of class 1 C .Then there exists a single triangle ( ) inscribed in the set S , in the sense that 1  x must be on the supporting line ( ) R is situated on at least one supporting line of the set convS (see for instance [3] pp. 6).We distinguish now the following two cases: 1) There is only one supporting line 1 L of the set convS going through the point p , i.e. the boundary { } convS ϑ is smooth in the point p .By Lemma 1 it follows that the convex hull convS is a strictly convex set and thereby we have Let us now suppose the point p S ∉ .From   H be the open halfplane generated by the line , L q q , which contains the point p and

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H the other open halfplane generated by the line ( ) , L q q .We have then evidently Then the point p must be an interior point of the convex hull convS .Let L be an arbitrary line such that p L ∈ .Then it is obvious that the line L intersects A similar result as that of Lemma 3 without the compactness of the set S but with the additional assumption of the connectedness of the set S was obtained by K. Juul in [7] , , a a a ∆ with ( ) ( ) and such that angle = such that triangle ( ) , , a a a ′ ′ ′ ∆ is homothetic to the triangle ( ) , , a a a ∆ .Because the angle .From the homothety of the triangles ( ) , , a a a ′ ′ ′ ∆ and ( ) , , a a a ∆ it follows then that { } { } { }

Conclusions
As we have seen condition ( 1) is used and is essential in the proofs of the Theorem 2, Theorem 3 and Theorem 4. We emit now the following: Conjecture: A compact set S in the Euclidean plane 2  R is the boundary of a smooth strictly convex set if and only if there is verified the condition: For every triangle ( ) P. Mani-Levitska cites in his survey [8] the papers [7] and [9] and says reffering to these, that he has not encountered extensions of these results to higher dimensions.We also don't know generalizations of our results to higher dimensions.
we denote the Euclidean distance of the points x and y and with ( ) , L x y the line determined by the points x and y.The diameter diamM of a set M is the circle and respectively the disc with center p and radius r .The distance ( ) , d p M between a point p and a set M in 2 open line segment with endpoints x and y , that is is in contradiction with property (2) of the set S .Thereby we must have p S ∈ .By an analog reasoning for the point q we can conclude that we have also: q S ∈ .Thus we have proved the existence of at least 2 different points of S on the supporting line ( ) , L p q of convS in contradiction to the property (2) of the set S .Lemma 2. The boundary { } convS ϑ of the convex hull of a compact set S in the Euclidean plane 2 R verifying the condition (2) from Theorem 4 is a subset of the set S, i.e. point from the boundary of the convex hull of the compact set S .Each boundary point of the compact convex set convS in 2 1 convS L p ∩ = and p S ∉ follows then 1 S L ∩ = ∅.Denote with o H the open halfplane generated by the line 1 L , which contains the set S .As S is a compact set we have then x S = ∈ > .Consider then in the open halfplane o H a line with the center p and the radius r we have: is the circle with center p and radius r .Let 1 and 2 t S ∈ in contradiction to the condition (3) of the set S .So we conclude that { } conv S S ϑ ⊂ .This inclusion together with the inclusion { } convS S

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angle bisector as the boundary angle of the cone C formed by the halflines 1L′ and 2 L′ with the vertex 1 a and such that the angle boundary angle of the cone C .By condition (1) there exists then three points , proved that the convex hull convS is a smooth strictly convex set.
∆ and inscribed in the set S i.e. such that 1 2 3 , , p p p S ′ ′ ′ ∈ .
is thereby supporting line for the compact set convS .Denote with To prove the only if part of the lemma let us consider a compact set S in the Euclidean plane 2 R , which verifies conditions (2) and (3).By Lemma 1 the convex hull convS of S is a strictly convex set.By Lemma 2 we have then for the set S the inclusion { } ∈. Therefore the point p must belong to the set S .
: Theorem 6.A connected set S in 2 R is a convex curve if and only if it ve- rifies condition (1) from Theorem 1. Proof of Theorem 4. For the proof of the if-part of the theorem let S be the boundary of a compact smooth strictly convex set in the Euclidean plane 2 R .It is then easy to ve-