Tangential projections and secant defective varieties

Going one step further in Zak’s classification of Scorza varieties with secant defect equal to one, we characterize the Veronese embedding of ${\mathrm{\P }}^n$ given by the complete linear system of quadrics and its smooth projections from a point as the only smooth irreducible complex and non-degenerate projective subvarieties of ${\mathrm{\P }}^N$ that can be projected isomorphically into ${\mathrm{\P }}^{2n}$ when $N\ge \left(\genfrac{}{}{0px}{}{n+2}{2}\right)-2$.