Talk:Convex curve/GA1

GA Review
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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. Lead: 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 —Kusma (talk) 14:12, 10 January 2023 (UTC)
 * 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 $$\liminf_{\delta\to0} \frac{\bigl|f(t)-f(t+\delta)\bigr|}{\delta}>0$$ at all parameter values $$t$$, which does not require $$f$$ 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 ≤ $\pi$ 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.
 * 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.
 * 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)