  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.
