Upper bound of the shortest vector coefficients and its applications

In this paper, we introduce a new notion normalized Gram matrix to provide new theorems of the geometry of lattices. Our theorems insist that the coefficients norm of the shortest vector of a given lattice is upper bounded by the eigenvalues of the normalized Gram matrix. Furthermore, we show that two toy applications of our theorems. First, using our upper bound, we can accelerate Kannan´s SVP algorithms for certain (very) special type of lattices. Second, we provide a criteria of the shortest vector for low dimensional lattices.