Note on affine Gagliardo-Nirenberg inequalities

This note proves sharp affine Gagliardo-Nirenberg inequalities which are stronger than all known sharp Euclidean Gagliardo-Nirenberg inequalities and imply the affine $L^p-$Sobolev inequalities. The logarithmic version of affine $L^p-$Sobolev inequalities is verified. Moreover, An alternative proof of the affine Moser-Trudinger and Morrey-Sobolev inequalities is given. The main tools are the equimeasurability of rearrangements and the strengthened version of the classical P'{o}lys-Szeg"{o} principle.