There is a long standing conjecture in Hamiltonian analysis which claims that there exist at least n geometrically distinct closed characteristics on every compact convex hypersurface in $\R^{2n}$ with n≥2. Besides many partial results, this conjecture has been only completely solved for n=2. In this paper, we give a confirmed answer to this conjecture for n=3. In order to prove this result, we establish first a new resonance identity for closed characteristics on every compact convex hypersurface $\Sg$ in $\R^{2n}$ when the number of geometrically distinct closed characteristics on $\Sg$ is finite. Then using this identity and earlier techniques of the index iteration theory, we prove the mentioned multiplicity result for $\R^6$. If there are exactly two geometrically distinct closed characteristics on a compact convex hypersuface in $\R^4$, we prove that both of them must be irrationally elliptic.