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 ${\bf R}^{2n}$ with $n\ge 2$ . Besides many partial results, this conjecture has been completely solved only for $n=2$ . In this article, 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 ${\Sigma}$ in ${\bf R}^{2n}$ when the number of geometrically distinct closed characteristics on ${\Sigma}$ is finite. Then, using this identity and earlier techniques of the index iteration theory, we prove the mentioned multiplicity result for ${\bf R}^6$ . If there are exactly two geometrically distinct closed characteristics on a compact convex hypersuface in ${\bf R}^4$ , we prove that both of them must be irrationally elliptic