The Dicks-Ivanov problem and the Hamidoune problem

We report on what we call the Hamidoune problem, inspired by a problem by Dicks and Ivanov. The problem asks if the inequality |A|+|B|−12|AB|−12|A⋅2B|⩽max{2,|gH|:H⩽G,g∈G,gH⊆A⋅2B} holds when A and B are finite subsets of a group G, each one with at least two elements, and A⋅2B denotes the set of elements which can be written in at least two different ways as a product of one element in A and one in B.