# What is the minimal angle between two LLL reduced vectors?

What is the minimal angle between two LLL reduced vectors? It seems it should be 60 degree as $|\mu_{i,j}| \leq \frac{1}{2}$. If we make the upper bound of $\mu_{i,j}$ by 1/3, can we get better reduction?

• How do you find a third vector in the two-dimensional lattice spanned by the two given vectors that fulfills your lower upper bound? Why should such a vector exist? Hint: 60 degree is optimal, as the three angles of a triangle sum up to 180 degree. – j.p. Nov 26 '17 at 16:49