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Exploiting the mass transportation method, we prove a dual principle whichimplies directly the sharp Gagliardo-Nirenberg trace inequalities which wasrecently proved by Bolley et al. [BCFGG17]. Moreover, we determine all optimalfunctions for these obtained sharp Gagliardo-Nirenberg trace inequalities. Thissettles a question left open in [BCFGG17]. Finally, we use the sharpGagliardo--Nirenberg trace inequality to establish their affine versions (i.e.,the sharp affine Gagliardo-Nirenberg trace inequalities) which generalize arecent result of De N\'apoli et al. [DeNapoli]. It was shown that the affineversions are stronger and imply the sharp Gagliardo-Nirenberg traceinequalities. We also determine all extremal functions for the sharp affineGagliardo--Nirenberg trace inequalities.

Sharp Gagliardo--Nirenberg trace inequalities via mass transportion method and their affine versions