# Talk:Convex curve

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

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)

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)
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)
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)
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)

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)

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

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 
  
    
      
        L
        ≡
        y
        =
        0
      
    
    {\displaystyle L\equiv y=0}
  
, 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.  — Preceding unsigned comment added by Laurent.Claessens (talk • contribs) 08:55, 13 April 2016 (UTC) 

Yes, this sounds correct — thanks for catching it. —David Eppstein (talk) 14:40, 13 April 2016 (UTC)

Once again in the proof of the "Parallel tangents". I guess that the hypothesis "closed" curve is missing. If not, the graph of the function 
  
    
      
        f
        (
        x
        )
        =
        
          x
          
            3
          
        
        −
        x
      
    
    {\displaystyle f(x)=x^{3}-x}
  
 with 
  
    
      
        x
        ∈
        [
        −
        5
        ,
        5
        ]
      
    
    {\displaystyle x\in [-5,5]}
  
 is a counter-example (even being compact). The point is that a closed curve can be seen as a map 
  
    
      
        R
        →
        
          R
          
            2
          
        
      
    
    {\displaystyle R\to R^{2}}
  
, so that every value of the parameter lies in the interior of the domain and the principle "maximum iif vanishing derivative" holds.
This being said, we should also ask for 
  
    
      
        
          C
          
            1
          
        
      
    
    {\displaystyle C^{1}}
  
. 
Laurent.Claessens (talk) 05:12, 15 April 2016 (UTC)

The Koch snowflake curve would seem to qualify as a convex curve according to the definition. (It has no tangent where the curve lies on both sides.) Is this intended?
Also it probably needs to be made clear that "one side of a line" is here intended to include the line itself. 
Martin Rattigan (talk) 16:22, 9 February 2017 (UTC)

Thanks to @David Eppstein, he did the revert of my commit. Because of his mathematical background I am certainly accepting I was wrong, consequently, I am having a really hard time to understand the difference between a convex curve and a convex function, even after reading the article multiple times. I admit the article states in its first line that it should not be confused with each other, however, if someone could explain the difference in the article a bit better I would be really thankful. Varagk (talk) 16:12, 28 October 2022 (UTC)

Additionally, I question if the picture of the parabola in the article is then also a suitable representation of the convex curve. Varagk (talk) 16:15, 28 October 2022 (UTC)
A function, in this context, maps each x coordinate to a single y-coordinate. A circle is a convex curve, but for some x-coordinates (the ones to the left or right of the circle) there is no corresponding y-coordinate, and for others (the ones between the leftmost and rightmost extreme points of the circle) there are two corresponding y-coordinates. The graph of a convex function is automatically convex curve, but not necessarily vice versa. So the parabola shown in the figure is a convex curve. —David Eppstein (talk) 16:31, 28 October 2022 (UTC)
@David Eppstein Thank you through your edits it's now really easy to understand and I don't feel completely clueless! Varagk (talk) 17:32, 28 October 2022 (UTC)
Wow thank you a lot for this fast reply and the great explanation. For my feeling the article would benefit a lot if you could include your second last sentence "The graph of a convex function is automatically convex curve, but not necessarily vice versa." in the article. However, I won't do this edit because I have the feeling you can decide better if and where to put this statement. But for me it would have made the understanding easier and I would have remembered that a function requires a x-coordinate to y-coordinate mapping. Thank you a lot! Varagk (talk) 16:40, 28 October 2022 (UTC)

This review is transcluded from Talk:Convex curve/GA1. The edit link for this section can be used to add comments to the review.
Reviewer: Kusma (talk · contribs) 10:52, 8 January 2023 (UTC)

Will take this one. Review to follow within a few days. —Kusma (talk) 10:52, 8 January 2023 (UTC)


### General comments and ticks
Overall well sourced and referenced and nicely illustrated with free images. Appears stable and neutral. Detailed comments to follow below. —Kusma (talk) 11:08, 10 January 2023 (UTC)


### Section by section review
Will do lead section last.

Definitions: Is there any disagreement on Archimedes and convexity so you need to mention Fenchel instead of stating this is Archimedes in wikivoice?
Ok, removed mention of Fenchel. —David Eppstein (talk) 00:39, 11 January 2023 (UTC)
Probably not something for you to do, but noting here anyway: Unfortunately plane curve muddies the waters by mixing the topological definition with that of an algebraic curve (the solution set of xy=1 is a plane algebraic curve that is not a topological curve).
I don't fully understand your definition of "regular". Do you have a derivative-free definition in mind when you say regular, meaning that the moving point never slows to a halt or reverses direction? You later have regular and has a derivative everywhere, but regular curve is only talking about differentiable curves.
I had in mind a definition in which a piecewise linear but nowhere-constant parameterization of a piecewise linear curve counts as regular, but where for instance a parameterization that is constant over some interval of the parameter space is not. I think the technical formulation of that is something like 
  
    
      
        
          lim inf
          
            δ
            →
            0
          
        
        
          
            
              
                
                  |
                
              
              f
              (
              t
              )
              −
              f
              (
              t
              +
              δ
              )
              
                
                  |
                
              
            
            δ
          
        
        >
        0
      
    
    {\displaystyle \liminf _{\delta \to 0}{\frac {{\bigl |}f(t)-f(t+\delta ){\bigr |}}{\delta }}>0}
  
 at all parameter values 
  
    
      
        t
      
    
    {\displaystyle t}
  
, which does not require 
  
    
      
        f
      
    
    {\displaystyle f}
  
 to be differentiable. That said, it appears that the article only uses regularity when talking about smooth curves, so there is no need for going into extra generality. I rearranged a bit to do this. —David Eppstein (talk) 00:39, 11 January 2023 (UTC)
Latecki is a slightly surprising choice for the "boundary of convex set is a convex curve" claim, especially as the chapter starts with "digital concepts". But the citation checks out (p. 42)
It can be surprisingly difficult to find sources that say certain obvious things explicitly. Another one that I am still hoping to track down a source for (but haven't, which is why it is not currently stated in the article) is that smooth arcs with non-negative curvature and with total curvature ≤ π must be convex. (If the total curvature exceeds π even by a little they can self-cross instead.) —David Eppstein (talk) 00:39, 11 January 2023 (UTC)
Intersection with lines: If you are bored, the characterisation of the intersection types could be nicely illustrated by an image. The "certain other linear spaces" bit is a bit mysterious without some example.
Added the real projective plane as an example, and an illustration. —David Eppstein (talk) 01:13, 11 January 2023 (UTC)
The link "locally equivalent" to local property is likely unhelpful to many readers, so the concept could be explained here a bit.
Ok, added an explanation. —David Eppstein (talk) 06:58, 11 January 2023 (UTC)
Length and area: link arc length instead of length?
Done. —David Eppstein (talk) 06:58, 11 January 2023 (UTC)
The Toponogov ref states the projection thing without invoking randomness, just as the average length of the projections, which seems an easier concept.
Ok, done. I agree that this is significantly less technical, generally a good thing in a Wikipedia article. It is also more vague, but I guess anyone who would ask "average over what distribution?" would be able to figure it out without having to be told. —David Eppstein (talk) 06:58, 11 January 2023 (UTC)
I wonder whether you could state more regularity here. By the Alexandrov theorem, the curve is almost everywhere twice differentiable, much better than rectifiability.
Interesting. Added, but not here: I added it at the end of the curvature section, since twice differentiability means it has a curvature at that point. —David Eppstein (talk) 07:14, 11 January 2023 (UTC)
Jarnik's bound cannot be improved: the source just says the bound is a "nearly best possible result" without making precise what that means. Is the exponent 1/3 the best possible? Is there an optimal constant? Or is this just about the leading term? Also, mention that this is the large-L asymptotics?
Both the leading constant and the exponent in the error term are best possible. Clarified to state that, and swapped out the source for one that says so more explicitly. The bound is valid for all L but clarified that it is accurate only for large L. —David Eppstein (talk) 01:38, 12 January 2023 (UTC)
Every curve has at most two supporting lines in each direction. can you clarify that we are looking at supporting lines of fixed direction, but at different points here? (The statement is "for every direction, there are at most two points such that there is a supporting line at that point in that direction", not "at every point there are at most two supporting lines")
I think the logic of this whole paragraph was hard to follow. The point was to prove a characterization of smooth closed convex curves in terms of the nonexistence of three parallel tangent lines. I rewrote the paragraph in an attempt to make its point more obvious. —David Eppstein (talk) 02:03, 12 January 2023 (UTC)
For strictly convex curves, although the curvature does not change sign, it may reach zero. perhaps add that simple closed curves with strictly positive / negative curvature are strictly convex?
Ok, done. I checked that this is also stated in the same source. —David Eppstein (talk) 08:14, 12 January 2023 (UTC)
Related shapes: I find it difficult to see the relevance of finite projective geometry here. Are you just disambiguating "oval" here or are you trying to say that this is a concept that is very similar in finite-set geometries and in the Euclidean plane?
Just disambiguating "oval". I reworded in an attempt to make that more clear. —David Eppstein (talk) 19:36, 12 January 2023 (UTC)
Mention Toeplitz' conjecture in this context to show that convex curves are quite special here? I'm not sure that the Akopyan-Avvakumov theorem is in the right section; this is a property of convex curves, not of some related shapes. Perhaps rename the section?
Ok, this one was a little complicated. I moved the "oval" material to a subsection of "definitions" on symmetry, renamed the section containing the inscribed quadrilateral material to "inscribed polygons", and expanded it a little. —David Eppstein (talk) 20:05, 12 January 2023 (UTC)
Anything about convex (hyper-)surfaces?
I added a see-also link to convex surface. However, it's currently just a redirect; we don't have a full article on that topic. —David Eppstein (talk) 20:12, 12 January 2023 (UTC)
Notes are quite helpful.
Lead: 

More or less says everything in the article, except perhaps the A-A theorem. The sentence Combinations of these properties have also been considered. could perhaps be dropped.
Ok, done.
A nice article about a basic topic, not much to complain about. More "advanced properties" like the A-A theorem or some "applications" would be nice, but not necessary for GA. Ping @David Eppstein: —Kusma (talk) 14:12, 10 January 2023 (UTC)

@Kusma: Ok, I think I have addressed everything. —David Eppstein (talk) 20:14, 12 January 2023 (UTC)
Indeed you have (moving around the ovals helps a lot). I'm happy now and will promote. —Kusma (talk) 21:24, 12 January 2023 (UTC)

The section Curvature contains this sentence:
"The total absolute curvature of a smooth convex curve, 
  
    
      
        ∫
        
          |
        
        κ
        (
        s
        )
        
          |
        
        d
        s
        ,
      
    
    {\displaystyle \int |\kappa (s)|ds,}
  
 is at most 
  
    
      
        2
        π
      
    
    {\displaystyle 2\pi }
  
. " 
My understanding of the word "smooth" is that it means infinitely differentiable (C∞).
But I have seen it used to mean merely continuously differentiable and other things as well.
The article would be improved if it stated exactly what it means by the word "smooth".
I hope someone knowledgeable about this subject can fix this ambiguity in the article. — Preceding unsigned comment added by 2601:200:c082:2ea0:8563:761:1843:919f (talk • contribs) 23:40, 16 March 2024 (UTC)

In general this is a can of worms. The "Background concepts" section already makes clear that this is ambiguous. The mathematics literature would be improved if it was commonplace to state exactly how smooth is smooth enough, but it often isn't. The short answer is often "smooth enough for the formula to be well defined", but often proofs of formulas go through higher levels of smoothness. I think for the four-vertex theorem there is an explicit statement that 
  
    
      
        
          C
          
            2
          
        
      
    
    {\displaystyle C^{2}}
  
 is enough in Thorbergsson and Umehara (1999), "A unified approach to the four vertex theorems II". I think the related tennis ball theorem can be proven more generally for 
  
    
      
        
          C
          
            1
          
        
      
    
    {\displaystyle C^{1}}
  
 curves [1] but although I have in mind a published proof that I think works with that assumption, the publication is not explicit on this point. To be more precise about how much smoothness is enough for the total absolute curvature bound, we would need to find a source that is similarly explicit about how much smoothness that bound needs. —David Eppstein (talk) 00:27, 17 March 2024 (UTC)