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In 2000, Paul R. Scott and Poh Way Awyong published the paper Inequalities for Convex Sets, which nicely collates the known results relating various natural geometric functionals (diameter, area, etc.) on convex planar sets.

Given that some of the inequalities in the paper cite results published just a few years prior (or in one case, 18 years later!), it seems that the state of knowledge about these inequalities is still evolving, so I would expect some new results to have arisen in the two decades since its original publication.

Which new results are known relating geometric functionals on planar convex sets? In particular, I would love to find an equivalent version of this paper with up-to-date information on the current state of knowledge, existing conjectures, etc., or a reliably-updated webpage tracking the same.

Two related extensions of the original paper I would be interested to see:

  • An extension of the table on relationships between pairs of functionals giving results for additional measurements, of which there are many possible choices: packing density, area of maximal inscribed triangle, etc.

  • An analogous collection of results for convex bodies in $\mathbb{R}^3$, though of course the number of natural functionals grows substantially and obtaining exact results is likely much harder. In the $\mathbb{R}^n$ case, this thread is a good start, though I'd be interested in seeing conjectural relationships as well.

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  • $\begingroup$ Previously on Math.StackExchange here, without any answers. $\endgroup$
    – RavenclawPrefect
    Commented Jan 16, 2021 at 20:21

1 Answer 1

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You can also consider "Minkowski differences" instead of "Minkowski sums":

  • Y. Martinez-Maure, Geometric study of Minkowski differences of plane convex bodies, Canadian Journal of Mathematics 58 (2006), 600-624.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/C13B5680AAEA98ABA76F1FB9956A968A/S0008414X00021969a.pdf/geometric_study_of_minkowski_differences_of_plane_convex_bodies.pdf

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  • $\begingroup$ 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! $\endgroup$
    – RavenclawPrefect
    Commented Jan 24, 2021 at 8:46
  • $\begingroup$ 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. $\endgroup$
    – Clement
    Commented Jan 24, 2021 at 9:46
  • $\begingroup$ 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. $\endgroup$
    – RavenclawPrefect
    Commented Jan 24, 2021 at 17:52
  • $\begingroup$ 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. $\endgroup$
    – Clement
    Commented Jan 24, 2021 at 22:14

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