# Inequality with volume in Euclidean Space

Given an Euclidean Space and vectors $a_1, a_2, \ldots a_k, b_1, b_2, \ldots b_n$.

Is it true that $V(a_1, a_2, \ldots , a_k, b_1, b_2, \ldots , b_n) \leq V(a_1, a_2, \ldots , a_k) \cdot V( b_1, b_2, \ldots , b_n)$ ?

EDIT: $V(a_1,a_2,…,a_k,b_1,b_2,…,b_n)$ is volume in k+n dimensional parallelogram, and the other V's are in k and n dimensional parallelogram, respectively.

• What do you mean by $V$? – PhoemueX Dec 30 '14 at 16:59
• Volume of parallelogram span by vectors. – Jerry Johanson Dec 30 '14 at 17:07
• Do you mean $V(a_1, a_2, \ldots , a_k, b_1, b_2, \ldots , b_n)$ is the "volume" (measure) in $k+n$ dimensional space, and the other $V$'s are in $k$ and $n$ dimensional space, respectively? – Rory Daulton Dec 30 '14 at 18:13
• Yes, you're right. – Jerry Johanson Dec 30 '14 at 18:23

The answer is positive and it follows from generalized Hadamard inequality for Gram determinant. The proof is in Chapter IX, $\S$5 of "The theory of matrices" by F. R. Gantmacher. There are also translations “Matrix theory”, 1 (1959) and “The theory of matrices” 1, (1977) published by Chelsea.