Timeline for What inequalities for convex sets are known since the work of Scott and Awyong?
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| Jan 24, 2021 at 22:14 | comment | added | Clement | I think that hedgehogs are more useful in higher dimensions. In the planar case, I know that (under regularity assumptions) the same author gave a geometrical interpretation of an upper bound of the isoperimetric deficit of convex curves in terms of signed area of their evolute (which of course is not convex). There are extensions to higher dimensions but I don't remember the details. I did not say that there are such results in this paper but only that it might be interesting to look in that direction. I am sorry if it is not relevant. | |
| Jan 24, 2021 at 17:52 | comment | added | RavenclawPrefect | I agree with all of those statements, but I'm still not sure how this paper relates to the parent question. Have these hedgehogs been used to prove novel inequalities between functionals of convex planar sets? I don't understand the relevance of hedgehogs to this question beyond the fact that they also have something to do with convex geometry. | |
| Jan 24, 2021 at 9:46 | comment | added | Clement | Minkowski addition is involved in certain proofs. The Minkowski sum of two convex planar sets is still convex, whereas their "Minkowski difference" is not. But, many notions extend to these "Minkowski differences" (the so-called hedgehogs) and quite a number of classical results find their counterparts. Of course, a few adaptations are necessary. In particular, volumes have to be replaced by their algebraic versions. For instance, the isoperimetric ineqality has a partial extension to this framework. | |
| Jan 24, 2021 at 8:46 | comment | added | RavenclawPrefect | I’m a little confused by this answer? The paper does mention convex planar sets, but I confess I don’t see the relevance to Scott and Awyong’s paper outside of a bit of Section 5, which proves some results about certain decidedly non-convex sets. I also don’t think the original post, or the literature it cites, are particularly focused on Minkowski summation? Apologies if I’ve missed something relevant in the paper or otherwise misunderstood your answer! | |
| Jan 24, 2021 at 8:40 | history | answered | Clement | CC BY-SA 4.0 |