Talk:Convex polygon

Strictly convex vs. convex polygon ??

I think that the only property that makes a convex polygon convex is all of the angles being less than or equal 180 degrees. That's all.--Luke Elms 20:23, 22 November 2006 (UTC)

I agree. "Equivalently, a polygon is strictly convex if every line segment between two nonadjacent vertices of the polygon is strictly interior to the polygon except at its endpoints" is clearly wrong - two opposite sides of a square can be connected at their borders, leading to a connection not strictly interior to the square, being on one of its sides. 80.219.170.151 (talk) 12:39, 22 May 2009 (UTC)

@Luke -The article lists different ways to interpret the definition of convex polygon. All interior angles being less than of equal to 180° is one of the way.

@80.219.170.151 -The sentence clearly mentions non-adjacent vertices. If two adjacent vertices are connected they will form a side of the polygon. Also vertices are not same as sides. -- Myth (Talk) 22:29, 27 June 2009 (UTC)

You're all wrong above. I have successfully completed graduate math courses that covered the subject.
The boundary of a polygon (all of its edges) is included in the polygon. We also need to be concerned about a definition of a convex polygon is useful.
Definition: A polygon is convex if and only if for any two points chosen in the polygon [including its edges, remember], the line segment connecting the two lies entirely in the the polygon.
One good thing about this definition is that it works for figures in the plane that are not polygons: in other words, they have some or all curved edges. For example, by using the above definition, all circles and all ellipses (for example) are also convex. Also, all actual triangles are convex, all rectangles are convex, and in fact, all parallelograms are convex. {squares and rhombuses included). Furthermore all trapezoids are convex, and all regular polygons are convex. 98.67.97.108 (talk) 13:51, 13 August 2010 (UTC)

I propose a change to the phrase "every line segment between two nonadjacent vertices of the polygon" that should cover the concerns above. Change to: "every pair of points within the polygon defines a line segment that does not pass outside the polygon" If I don't hear any objection to this after a reasonable amount of time, I'll make the change. Hubby2debbie (talk) 14:15, 2 January 2014 (UTC)

Less than 180 degrees?

Might be a bit pedantic but this is math, it must be: Shouldn't it be "less or equal than 180 degrees"? Take the first image for example. It has 5 vertices. But now imagine there is a 6th vertex on the line between any two vertices. The shape of the polygon does not change. However, according to the definition in this article, it is not a convex polygon anymore as the angle between the 5th and the 6th vertex is 180°. This must clearly be a mistake as the changed polygon still inherits all the properties of the original one.

Nope - all of the vertices with measure of 180 degrees are uninteresting and the contribute noting to the subject. So "less than 180 degrees is correct".

98.67.97.108 (talk) 14:05, 13 August 2010 (UTC)

Furthermore: This one "Every line segment between two vertices remains inside or on the boundary of the polygon." is not needed. The first point is enough for the definition. The second point derives from the first. OK, probably the first point would derive from the second point, too. But I think it is important to say that to define a convex polygon, one of these two points is enough.

--89.53.92.77 (talk) 13:54, 2 October 2009 (UTC)

Can a vertex have a 180-degree angle, or is that not allowed under its definition? Because any polygon would have an infinite number of 180-degree vertices. I'm not a mathematician, but I'm sure this is the reason the article reads the way it does (I may even have made that change!) - Special-T (talk) 12:57, 3 October 2009 (UTC)

Yes, you are right. Every nondegenetrate polygon has an infinite number of 180-degree vertices. (A degenerate polygon is one that has been shrunken down to a single point. Clearly, that does not have any vertices at all.) In fact, there is an uncountable infinity of 180-degree vertices in a polygon, so 180-degree vertices are completely unintersting and they don't need to be included in the definition. So, the actual vertices of a convex polygon all measure less than 180 degrees (or in other words π radians).

98.67.97.108 (talk)

A polygon does not have infinitely many vertices with interior 180-degree angles. A polygon cannot have infinitely many vertices at all. It is possible for a polygon to have some vertices with interior 180-degree angles. For example, the polygon with vertices (0, 0), (1, 0), (2, 0), (1, 1) in order has a 180-degree angle at (1, 0), while the polygon with vertices (0, 0), (2, 0), (1, 1) in order (although it "looks" the same and has the same interior) does not. Therefore, "each interior angle less than 180 degrees" is not equivalent to the convexity of the interior of the polygon, and the phrase "less than or equal to" must be used in place of "less than." It is important to include the possibility of these vertices even though they are trivial. Trivialsegfault (talk) 01:03, 3 May 2011 (UTC)

Split

I just split the content from convex and concave polygons about concave polygons to concave polygon and renamed this article to convex polygon. — Martin (MSGJ · talk) 13:03, 29 April 2015 (UTC)

Determining convexity

I suggest this addition to the article: what is the most efficient way of determining whether a polygon is convex, given the coordinates of its vertices? Loraof (talk) 21:28, 14 October 2015 (UTC)

The test that an interior angle has less than 180° is the computation of the sign of a cross product. Thus the test of convexity has complexity O(n). This cannot be improved, as one has to consider all the vertices for testing convexity.

The computation of the convex hull of a finite set is a more interesting related problem. I have just added a link to convex hull in the body of the article. Now, the question becomes: is the computation of the complex hull of a finite set in the plane sufficiently detailed in convex hull, or must we add, here, a detailed section on this problem? D.Lazard (talk) 07:12, 15 October 2015 (UTC)

Convex hull#Computation of convex hulls links to Convex hull algorithms, which gives a good discussion.

You added to this article

• The polygon is the boundary of the convex hull of its edges.
• The polygon is the boundary of the convex hull of a finite points of the plane.

Both of these assume that the polygon does not include its interior, but conventionally it does—see Polygon. Also, when the second one is corrected for this, it more or less duplicates the third item in the list of additional properties:

• Krein–Milman theorem: A convex polygon is the convex hull of its vertices.

So I'm going to revise the first added sentence and delete the second as redundant.

Loraof (talk) 14:01, 15 October 2015 (UTC)