Companions to the Brunn–Minkowski inequality

In this paper some improvements of the Brunn–Minkowski inequality are discussed by using the negativity of certain functional related to this inequality. To do so, a majorization preorder with respect to Minkowski sum is introduced and applied. A corresponding Hardy–Littlewood–Pólya–Karamata (HLPK) type theorem for the functional is proven. In consequence, a refinement of the Brunn–Minkowski inequality is established.