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Book "Treatise of Avalysis" Vol. IV DIEUDONNE has nothing common with book of Girkin "Spectral Theory of Random Matrices"/ It look like error link of Google. Jumpow (talk) 15:04, 23 February 2015 (UTC)Jumpow[reply]

Different passages in the article either require or don't require a convex curve to be closed.

From the lead:

A convex curve is a curve ... which lies on one side of each of its tangent lines.

From "Definition by supporting lines":

A plane curve is called convex if it lies on one side of each of its tangent lines.

From "Definition by convex sets":

A convex curve may be defined as the boundary of a convex set....[or] a subset of the boundary of a convex set.

From "Properties":

Every convex curve has a well-defined finite length.

The first two quotes imply that a parabola is a convex curve, while the last two imply that it is not. If standard terminology requires it to be a closed curve (or subset thereof), the first two quotes should be modified to reflect that. On the other hand, if the term is used both ways, with and without a restriction that the curve be closed, then this should be explicitly mentioned. Thanks. Loraof (talk) 16:14, 28 May 2015 (UTC)[reply]

The third quote does not necessarily contradict the first two. E.g, a parabola can also be seen as a boundary of a convex (unbounded) set. I am not sure about the 4th quote. --Erel Segal (talk) 19:04, 28 May 2015 (UTC)[reply]

But the parabola is not a closed curve, so the claim that the "boundary of convex set" definition implies that the curve is closed appears to be incorrect. As another example: an open semicircle (i.e. one that is missing its two endpoints) would seem to satisfy the definition by supporting lines, but is not the boundary of a convex set (instead it obeys the "subset of the boundary" definition). The statement of the four-vertex theorem is also incorrect; it requires smoothness. —David Eppstein (talk) 19:34, 28 May 2015 (UTC)[reply]

Right, Erel, the third quote above permits parabolas; unfortunately I left out a key part of the passage. The complete version of the third quote is

A convex curve may be defined as the boundary of a convex set in the Euclidean plane. This means that a convex curve is always closed (i.e. has no endpoints). Sometimes, a looser definition is used, in which a convex curve is a curve that forms a subset of the boundary of a convex set. For this variation, a convex curve may have endpoints.

As David points out, the second sentence here does not logically follow from the first one.

I disagree with David about his example the open semicircle--I think it is the boundary of a convex set, namely an open half-disk. Loraof (talk) 16:20, 29 May 2015 (UTC) Strike that-- of course it's a subset of the boundary. Loraof (talk) 16:37, 29 May 2015 (UTC)[reply]

Also, the article four-vertex theorem defines a convex curve as one with strictly positive curvature. Modifying this to say non-negative curvature (to allow for the non-strict case) would seem to me to be another good definition (equivalent I think to the one about tangent lines) which does not appear in this article. Loraof (talk) 16:32, 29 May 2015 (UTC) Strike that too--it's in there toward the bottom. Loraof (talk) 16:55, 29 May 2015 (UTC)[reply]

I would suggest that the following two related issues be discussed in this article:

1. Given the equation of an algebraic plane curve (or perhaps more specifically a closed one), how does one determine whether it is convex?

2. Given the vertex coordinates of a polygon, what is the most efficient way to determine if it is convex?

Loraof (talk) 20:43, 14 October 2015 (UTC)[reply]

I don't know about the algebraic version of the question, but for point sets, if you're just given the vertices in arbitrary order, you should compute their convex hull. If you're given a sequence of vertices that is intended to be their cyclic sequence as vertices of a convex polygon, then you can verify that it really is convex by checking that all consecutive triples are consistently oriented and that the turning number is one. See e.g. Schorn, Peter; Fisher, Frederick (1994), "I.2 Testing the convexity of a polygon", in Heckbert, Paul S. (ed.), Graphics Gems IV, Morgan Kaufman (Academic Press), pp. 7–15, ISBN 9780123361554. —David Eppstein (talk) 22:17, 14 October 2015 (UTC)[reply]

Thanks! I'll take a look at that. Loraof (talk) 23:48, 14 October 2015 (UTC)[reply]

Also there's a trick for discretizing the turning number into multiples of π/2 so that you can use integer arithmetic in computing it. —David Eppstein (talk) 01:09, 15 October 2015 (UTC)[reply]

I think that I found an error in the proof of the "Parallel tangents". It is said that q1 is the farthest point from p. I guess that q1 has to be the farthest point from L.

Actually taking an axe system in which p=(0,0) and , it is clear that the farthest point from L is a point on which the derivative of $C_y$ vanishes. This is also the meaning of the Hint on page 6 here here.

If no reaction, I'll do the change.