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In the first part of this paper we obtain a sharp inequality for the Strichartz norm $\\|u(t,x)\\|_{L^{2k}_tL^{2k}_x(\\R \\times\\R^n)}$, where $k\\in \\Z$, $k \\geq 2$ and $(n,k) \\neq (1,2)$, that admits only Gaussian maximizers. As corollaries we obtain sharp forms of the classical Strichartz inequalities in low dimensions (works of Foschi and Hundertmark - Zharnitsky) and also sharp forms of some Sobolev-Strichartz inequalities. 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