# What inequalities for convex sets are known since the work of Scott and Awyong?

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.

• Previously on Math.StackExchange here, without any answers. – RavenclawPrefect Jan 16 at 20:21