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Multiple brake orbits on compact convex symmetric reversible hypersurfaces in $\R^{2n}$

In this paper, we prove that there exist at least $[\frac{n+1}{2}]+1$ geometrically distinct brake orbits on every $C^2$ compact convex symmetric hypersurface $\Sg$ in $\R^{2n}$ for $n\ge 2$ satisfying the reversible condition $N\Sg=\Sg$ with $N=\diag (-I_n,I_n)$. As a consequence, we show that there exist at least $[\frac{n+1}{2}]+1$ geometrically distinct brake orbits in every bounded convex symmetric domain in $\R^{n}$ with $n\ge 2$ which gives a positive answer to the Seifert conjecture of 1948 in the symmetric case for $n=3$. As an application, for $n=4$ and 5, we prove that if there are exactly $n$ geometrically distinct closed characteristics on $\Sg$, then all of them are symmetric brake orbits after suitable time translation.