Bonnesen (1929) improved the isoperimetric inequality
for an arbitrary planar convex body K, stating that
Topics include a review of preliminaries such as continuous and Holder continuous functions, Sobolev spaces and convex analysis; classical methods such as Euler-Lagrange equations; direct methods such as the Dirichlet integral; regularity, such as the one-dimensinal case; minimal surfaces such as in the Douglas- Courant-Tonelli method; and isoperimetric inequality
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In addition, there is an isoperimetric inequality that establishes a bound on the area in terms of the length.
We can estimate most of the density by Wirtinger's inequality; we use the isoperimetric inequality to estimate the remainder.
This fact, combined with the isoperimetric inequality, gives that M(T L (([[omega].sub.1] [intersection] [[omega].sub.2].sup.c] [intersection] B(p,[epsilon]))) [less than or equal to] CM([alpha]T L ([K.sup.c] [intersection] B(p,2[[epsilon))).sup.2] = 0([epsilon]).
The classical isoperimetric inequality states that if K is convex body (see, e.g., [50, p.382]), then
The Orlicz isoperimetric inequality is established in Section 5.
Ball, Volume ratios and a reverse isoperimetric inequality