Every locally maximal product-free set S in a finite group G satisfies G=S∪SS∪S−1S∪SS−1∪S−−√, where SS={xy∣x,y∈S}, S−1S={x−1y∣x,y∈S}, SS−1={xy−1∣x,y∈S} and S−−√={x∈G∣x2∈S}. To better understand sets, Bertram asked whether every abelian satisfy |S−−√|≤2|S|. This question was recently answered the negation by current author. Here, we improve some results on structures sizes of groups terms their ...

The goal of this paper is to study geometric and extremal properties the convex body BF(M), which unit ball Lipschitz-free Banach space associated with a finite metric M. We investigate ℓ1 ℓ∞-sums, in particular we characterize spaces such that BF(M) Hanner polytope. also whose are isometric. discuss extreme volume product P(M)=|BF(M)|⋅|BF(M)∘|, when number elements M fixed. show if P(M) maxima...