On Two Conjectures concerning Convex Curves
نویسنده
چکیده
In this paper we recall two basic conjectures on the developables of convex projective curves, prove one of them and disprove the other in the first nontrivial case of curves in RP . Namely, we show that i) the tangent developable of any convex curve in RP 3 has degree 4 and ii) construct an example of 4 tangent lines to a convex curve in RP 3 such that no real line intersects all four of them. The question (discussed in [EG1] and [So4]) whether the second conjecture is true in the special case of rational normal curves still remains open. §
منابع مشابه
¡div class=”moz-text-flowed” style=”font-family: -moz-fixed”¿ ON TWO CONJECTURES CONCERNING CONVEX CURVES
In this paper we recall two basic conjectures on the developables of convex projective curves, prove one of them and disprove the other in the first nontrivial case of curves in RP . Namely, we show that i) the tangent developable of any convex curve in RP 3 has degree 4 and ii) construct an example of 4 tangent lines to a convex curve in RP 3 such that no real line intersects all four of them....
متن کاملTwo Conjectures on Convex Curves
Department of Mathematics, University of Oil and Gas (Gubkin) Moscow 117036, Russia [email protected] Department of Mathematics, University of Stockholm S-10691, Sweden, [email protected] Abstra t. In this paper we recall two basic conjectures on the developables of convex projective curves, prove one of them and disprove the other in the first nontrivial case of curves in RP3. Namely...
متن کاملAnalysis of Venn Diagrams Using Cycles in Graphs
This paper is the last in a series by the authors on the use of graph theory to analyze Venn diagrams on few curves (see [1,2,6,7]). We complete the construction (and hence the enumeration) of spherical Venn diagrams on five curves, which yields additional results about conjectures of Grünbaum concerning which Venn diagrams are convex, which are exposed, and which can be drawn with congruent el...
متن کاملar X iv : m at h / 02 08 21 8 v 1 [ m at h . A G ] 2 8 A ug 2 00 2 TWO CONJECTURES ON CONVEX CURVES
In this paper we recall two basic conjectures on the developables of convex projective curves, prove one of them and disprove the other in the first nontrivial case of curves in RP 3. Namely, we show i) that the tangent developable of any convex curve in RP 3 has degree 4 and ii) construct an example of 4 tangent lines to a convex curve in RP 3 such that no real line intersects all four of them...
متن کاملOn Kalai's Conjectures Concerning Centrally Symmetric Polytopes
In 1989 Kalai stated the three conjectures A, B, C of increasing strength concerning face numbers of centrally symmetric convex polytopes. The weakest conjecture,A, became known as the “3-conjecture”. It is well-known that the three conjectures hold in dimensions d ≤ 3. We show that in dimension 4 only conjectures A and B are valid, while conjecture C fails. Furthermore, we show that both conje...
متن کامل