Approximate convexity and an edge-isoperimetric estimate

An Error Estimate for the Isoperimetric Deficit

A four part dissection and rearrangement provides a new proof of the isoperimetric inequality in the plane as well as a new approach to Bonnesen-type error estimates for the isoperimetric deficit of compact convex sets and of star bodies that are centrally symmetric with respect to the origin. An isoperimetric inequality in R bounds the area (or related functional) of a compact set by some func...

Isoperimetric Flow and Convexity of H-graphs

In this paper we consider a “flow” of nonparametric solutions of the volume constrained Plateau problem with respect to a convex planar curve. Existence and regularity is obtained from standard elliptic theory, and convexity results for small volumes are obtained as an immediate consequence. Finally, the regularity is applied to show a strong stability condition (Theorem 8) for all volumes cons...

AN APPROXIMATE ISOPERIMETRIC INEQUALITY FOR r-SETS

We prove a vertex-isoperimetric inequality for [n], the set of all r-element subsets of {1, 2, . . . , n}, where x, y ∈ [n] are adjacent if |x∆y| = 2. Namely, if A ⊂ [n] with |A| = α ( n r ) , then its vertex-boundary b(A) satisfies |b(A)| ≥ c √ n r(n− r)α(1− α) ( n r )

Remarks on an Edge Isoperimetric Problem

Among all collections of a given number of k-element subsets of an n-element groundset find a collection which maximizes the number of pairs of subsets which intersect in k − 1 elements. This problem was solved for k = 2 by Ahlswede and Katona, and is open for k > 2. We survey some linear algebra approaches which yield to estimations for the maximum number of pairs, and we present a new and sho...

Approximate Convexity and Submonotonicity in Locally Convex Spaces