Talk:Convex curve/GA1

GA Review

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Reviewer: Kusma (talk · contribs) 10:52, 8 January 2023 (UTC)[reply]

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

General comments and ticks

Good Article review progress box

Criteria: 1a. prose () 1b. MoS () 2a. ref layout () 2b. cites WP:RS () 2c. no WP:OR () 2d. no WP:CV ()

3a. broadness () 3b. focus () 4. neutral () 5. stable () 6a. free or tagged images () 6b. pics relevant ()

Note: this represents where the article stands relative to the Good Article criteria. Criteria marked are unassessed

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)[reply]

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?
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.
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)
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.
The link "locally equivalent" to local property is likely unhelpful to many readers, so the concept could be explained here a bit.
Length and area: link arc length instead of length?
The Toponogov ref states the projection thing without invoking randomness, just as the average length of the projections, which seems an easier concept.
I wonder whether you could state more regularity here. By the Alexandrov theorem, the curve is almost everywhere twice differentiable, much better than rectifiability.
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?
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")
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?
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?
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?
Anything about convex (hyper-)surfaces?
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.

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)[reply]