One might notice a few similarities between VG and the isodiametric function considered by Gersten (see ).
A refinement of the simple connectivity at infinity of groups
Recall that for any measurable set O the isodiametric inequality yields s ≤ diam(O)/2.
Aperiodic fractional obstacle problems
The isodiametric inequality is derived from the isoperimetric inequality trough a variational principle, establishing that balls maximize the perimeter among convex sets with fixed diameter.
Quantitative stability in the isodiametric inequality via the isoperimetric inequality
This principle brings also quantitative improvements to the isodiametric inequality, shown to be sharp by explicit nearly optimal sets.
Quantitative stability in the isodiametric inequality via the isoperimetric inequality
By the above argument, we have Hn−1 (Fν ) = ωn−1 for a.e. ν ∈ ∂B , i.e., Fν is optimal in the isodiametric inequality in Rn−1 , and thus it is an (n − 1)-dimensional unit disk in ν ⊥ .
Quantitative stability in the isodiametric inequality via the isoperimetric inequality
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