the intutionistic meaning of the Lovász condition in the LLL algorithm

Question

I'm reading materials about the LLL algorithm. The LLL algorithm finds an LLL reduced basis for a lattice, where the LLL reduced is defined as in this picture.

I want to know the intuitionistic meaning of the second Lovász condition.

Can anyone help me? Thank you.

Answer 1

Quoted from the slides Joseph H. Silverman - An Introduction to the Theory of Lattices and Applications to Cryptography

Lovasz condition:

$$ \|\mathbf{v}_{i+1}^*\| \ge \sqrt{\frac{3}{4} - \frac{\left | \mathbf{v}_{i+1} \cdot \mathbf{v}_i^* \right |^2 }{\|\mathbf{v}_i^*\|^2}} \|\mathbf{v}_i^*\|$$

What a mess, right! But geometrically, the Lovasz Condition says that:

Projection of $\mathbf{v_{i+1}}$ onto Span$(\mathbf{v}_1,\dots, \mathbf{v}_{i-1})^{\bot} \ge \frac{3}{4} \cdot$ Projection of $\mathbf{v}_i$ onto Span$(\mathbf{v}_1,\dots, \mathbf{v}_{i-1})^{\bot}$

Basically it means that every following vector has a lower bound when it comes to the size of the vector.