_{1}

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

The reader unfamiliar with the theory of convex sets is referred to the books [

] x , y [ = conv { x , y } \ { x , y } . For 3 nonlinear points x , y and z in R 2 we denote with max ∠ ( x , y , z ) the maximum angle of the triangle Δ ( x , y , z ) . A convex curve is a connected subset of the boundary of a convex set.

In the chapter 8 of the book [

Theorem 1. A plane set S fulfils

1) ∀ x , y , z ∈ S : S ∩ int { conv { x , y , z } } = ∅ , if and only if S is either a subset of the boundary of a convex set, or an X -set, that is a set { x 1 , x 2 , x 3 , x 4 , x 5 } with ] x 1 , x 2 [ ∩ ] x 3 , x 4 [ = { x 5 } .

A survey of different characterizations of convex sets is given in the paper [

In the years 1978 [

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) ∀ x , y , z ∈ S : S ∩ int { conv { x , y , z } } = ∅ ,

2) For every triangle Δ ( p 1 , p 2 , p 3 ) in R 2 there is one and only one triangle Δ ( p ′ 1 , p ′ 2 , p ′ 3 ) homothetic to the triangle Δ ( p 1 , p 2 , p 3 ) inscribed in the set S , i.e. such that p ′ 1 , p ′ 2 , p ′ 3 ∈ S .

Theorem 3. A plane compact set S is the boundary of a smooth strictly convex set if and only if the following two conditions hold:

1) For every p ∈ S and every ϵ > 0 there is a positive number δ ( p , ϵ ) such that for every triplet of nonlinear points r , s , t in S ∩ int { D ( p , δ ( p , ϵ ) ) } we have max ∠ ( r , s , t ) > π − ϵ .

2) For every triangle Δ ( p 1 , p 2 , p 3 ) in R 2 there is one and only one triangle Δ ( p ′ 1 , p ′ 2 , p ′ 3 ) , homothetic to the triangle Δ ( p 1 , p 2 , p 3 ) inscribed in the set S , i.e. such that p ′ 1 , p ′ 2 , p ′ 3 ∈ S .

The main result of this paper is Theorem 4 giving another characterization of the boundaries of smooth strictly convex sets in the Euclidean plane R 2 which uses also condition (2) of the Theorem 2 and Theorem 3.

Theorem 4. A compact set S in the Euclidean plane R 2 is the boundary of a smooth strictly convex set if and only if there are verified the following three conditions:

1) For every triangle Δ ( p 1 , p 2 , p 3 ) in R 2 there is one and only one triangle Δ ( p ′ 1 , p ′ 2 , p ′ 3 ) homothetic to the triangle Δ ( p 1 , p 2 , p 3 ) inscribed in the set S , i.e. such that p ′ 1 , p ′ 2 , p ′ 3 ∈ S .

2) For any two distinct points p ∈ S and q ∈ S there are at least two points t 1 and t 2 such that t 1 ∈ S ∩ H 1 and t 2 ∈ S ∩ H 2 , where H 1 and H 2 are the two open halfplanes generated in R 2 by the line L ( p , q ) .

3) The set S does not contain three collinear points.

For the proof of Theorem 4 we need the following theorem from the paper [

Theorem 5. Let Δ ( a , b , c ) be a triangle in the Euclidean plane R 2 . Suppose that S is a strictly convex closed arc of class C 1 . Then there exists a single triangle Δ ( a 1 , b 1 , c 1 ) homothetic to the triangle Δ ( a , b , c ) inscribed in the set S , in the sense that a 1 , b 1 , c 1 ∈ S .

Lemma 1. The convex hull conv S of a compact set S in the Euclidean plane R 2 verifying the condition (2) from Theorem 4 is a strictly convex set.

Proof. Let us suppose the contrary. Then there are two distinct points

p , q ∈ ϑ { conv S } such that the line segment conv { p , q } ⊂ conv S . The convex hull of a compact set is also a compact set (see [

If the point p can be expressed only as a convex combination of three (and not of fewer) points x 1 , x 2 , x 3 of S then it follows that we must have

p ∈ int { conv { x 1 , x 2 , x 3 } } ⊂ int { conv S } in contradiction to the fact that

p ∈ ϑ { conv S } .

If the point p can be expressed only as a convex combination of 2 (and not of fewer) points of S , there are x 1 ∈ S and x 2 ∈ S such that

p ∈ conv { x 1 , x 2 } ⊂ conv S ⊂ c l { H 1 } . Then the points x 1 and x 2 must be on the supporting line L ( p , q ) . As H 2 ∩ conv S = ∅ , this 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 conv S in contradiction to the property (2) of the set S .

Lemma 2. The boundary ϑ { conv S } of the convex hull of a compact set S in the Euclidean plane R 2 verifying the condition (2) from Theorem 4 is a subset of the set S, i.e. ϑ { conv S } ⊂ S .

Proof. Let p ∈ ϑ { conv S } be an arbitrary point from the boundary of the convex hull of the compact set S . Each boundary point of the compact convex set conv S in R 2 is situated on at least one supporting line of the set conv S (see for instance [

1) There is only one supporting line L 1 of the set conv S going through the point p , i.e. the boundary ϑ { conv S } is smooth in the point p . By Lemma 1 it follows that the convex hull conv S is a strictly convex set and thereby we have conv S ∩ L 1 = p .

Let us now suppose the point p ∉ S . From conv S ∩ L 1 = p and p ∉ S follows then S ∩ L 1 = ∅ . Denote with H o the open halfplane generated by the line L 1 , which contains the set S . As S is a compact set we have then

r = min { d ( x , L 1 ) : x ∈ S } > 0 . Consider then in the open halfplane H o a line L ′ 1 parallel to the line L 1 at distance r to the line L 1 . Denote with H ′ o the open halfplane generated by the line L ′ 1 and such that H ′ o ⊂ H o . It is evident that p ∉ c l { H ′ o } . From the definition of the constant, r folows S ⊂ c l { H ′ o } and ϑ { conv S } ⊂ c l { H ′ o } in contradiction to p ∈ ϑ { conv S } . Thereby our supposition p ∉ S is false, i.e. we must have p ∈ S .

2) There are two supporting lines L 1 and L 2 of the set conv S going through the point p . Denote then with L ′ 1 and L ′ 2 the two halflines with endpoint p of the line L 1 and respectively L 2 such that

conv S ⊂ conv { L ′ 1 ∪ L ′ 2 } .

Let us suppose that p ∉ S . From the compactness of S follows then the existence of a real number r > 0 such that for the disc D ( p , r ) with the center p and the radius r we have: D ( p , r ) ∩ S = ∅ . Consider then the points q 1 = C ( p , r ) ∩ L ′ 1 and q 2 = C ( p , r ) ∩ L ′ 2 , where C ( p , r ) is the circle with center p and radius r . Let H 1 be the open halfplane generated by the line L ( q 1 , q 2 ) , which contains the point p and H 2 the other open halfplane generated by the line L ( q 1 , q 2 ) . We have then evidently S ∩ c l H 1 = ∅ and thereby S ⊂ H 2 . From the inclusion S ⊂ H 2 it follows also that conv S ⊂ H 2 . As ϑ { conv S } ⊂ S we have also: ϑ { conv S } ⊂ H 2 in contradiction to our supposition p ∈ ϑ { conv S } . Therefore the point p must belong to the set S .

So we have proved in both cases (1) and (2) that p ∈ ϑ { conv S } implies

p ∈ S , i.e. ϑ { conv S } ⊂ S .

A characterization of compact sets S in the Euclidean plane R 2 for which we have S = ϑ { conv S } is given in the following:

Lemma 3. A compact set S in the Euclidean plane R 2 has a strictly convex hull and coincides with the boundary of its convex hull ϑ { conv S } if and only if there are verified the conditions (2) and (3).

Proof. Let S be a compact set in the Euclidean plane R 2 , which has a strictly convex hull conv S and such that S = ϑ { conv S } . Consider then two arbitrary points p 1 and p 2 of the set S and the two open halfplanes generated by the line L ( p 1 , p 2 ) in R 2 . Because S has a strictly convex hull it is then evident that we have verified condition (2) and (3).

To prove the only if part of the lemma let us consider a compact set S in the Euclidean plane R 2 , which verifies conditions (2) and (3). By Lemma 1 the convex hull conv S of S is a strictly convex set. By Lemma 2 we have then for the set S the inclusion ϑ { conv S } ⊂ S . Let us now suppose that we would have S ⊂ ϑ { conv S } , i.e. there is a point p ∈ S such that p ∉ ϑ { conv S } . Then the point p must be an interior point of the convex hull conv S . Let L be an arbitrary line such that p ∈ L . Then it is obvious that the line L intersects ϑ { conv S } in two different points t 1 and t 2 such that p ∈ conv { t 1 , t 2 } . From ϑ { conv S } ⊂ S it follows that t 1 ∈ S and t 2 ∈ S in contradiction to the condition (3) of the set S . So we conclude that S ⊂ ϑ { conv S } . This inclusion together with the inclusion ϑ { conv S } ⊂ S gives then S ⊂ ϑ { conv S } .

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 [

Theorem 6. A connected set S in R 2 is a convex curve if and only if it verifies 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 R 2 . It is then easy to verify conditions (2) and (3) for the set S . Condition (1) follows immediately from Theorem 5.

For the proof of the “only if”―part of the theorem let S be a compact set in the Euclidean plane R 2 , which verifies conditions (1), (2) and (3). By Lemma 3 it follows that the convex hull conv S of the set S is strictly convex and that S = ϑ { conv S } .

It remains to prove that conv S is also a smooth set. Let us assume the contrary: there is a point a 1 ∈ ϑ { conv S } , which is not a smooth point of the boundary of S , i.e. there exist two supporting lines L 1 and L 2 for the set conv S at the point a 1 . For i ∈ { 1 , 2 } denote with H i the closed half-plane generated by the supporting line L i , which contains the set S . Denote with C the convex cone C = H 1 ∩ H 2 . We have then evidently the inclusions: S ⊂ C and conv S ⊂ C . As conv S is a strictly convex set we have also the inclusion S \ a 1 ⊂ int C . For i ∈ { 1 , 2 } denote with L ′ i the closed halfline of the line L i with origin a 1 such that L ′ i ⊂ ϑ C . Consider then the isosceles triangle

Δ ( a 1 , a 2 , a 3 ) with d ( a 1 , a 2 ) = d ( a 1 , a 3 ) and such that angle ∠ a 2 a 1 a 3 has the same angle bisector as the boundary angle of the cone C formed by the halflines L ′ 1 and L ′ 2 with the vertex a 1 and such that the angle ∠ a 2 a 1 a 3 is greater than the boundary angle of the cone C . By condition (1) there exists then three points a ′ i ∈ S , i = 1 , 2 , 3 such that triangle Δ ( a ′ 1 , a ′ 2 , a ′ 3 ) is homothetic to the triangle Δ ( a 1 , a 2 , a 3 ) . Because the angle ∠ a 2 a 1 a 3 is greater than the boundary angle of the cone C the point a ′ 1 cannot coincide with the point a 1 . From this fact and the inclusion S \ a 1 ⊂ int C we can conclude that we have: a ′ i ∈ int C for i = 1 , 2 , 3 . From the homothety of the triangles

Δ ( a ′ 1 , a ′ 2 , a ′ 3 ) and Δ ( a 1 , a 2 , a 3 ) it follows then that

a ′ 1 ∈ int { conv { a 1 , a ′ 2 , a ′ 3 } } ⊂ int { conv S } in contradiction to a ′ 1 ∈ S = ϑ { conv S } . So we have proved that the convex hull conv S is a smooth strictly convex set.

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 R 2 is the boundary of a smooth strictly convex set if and only if there is verified the condition:

For every triangle Δ ( p 1 , p 2 , p 3 ) in R 2 there is one and only one triangle Δ ( p ′ 1 , p ′ 2 , p ′ 3 ) homothetic to the triangle Δ ( p 1 , p 2 , p 3 ) and inscribed in the set S i.e. such that p ′ 1 , p ′ 2 , p ′ 3 ∈ S .

P. Mani-Levitska cites in his survey [

The author is grateful to the referees for the helpful comments.

Kramer, H. (2017) Boundaries of Smooth Strictly Convex Sets in the Euclidean Plane R^{2}. Open Journal of Discrete Mathematics, 7, 71-76. https://doi.org/10.4236/ojdm.2017.72008