  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 $n\ge 2$. 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.
