On an inequality related to the volume of a parallelepiped

Abstract

The problem of establishing an upper bound for the volume of a parallelepiped is considered by utilizing an original approach involving a skew-symmetric matrix of order four (along with its Moore–Penrose inverse). It is shown that the commonly known inequality characterizing the bound can be virtually sharpened. Similarly, a sharpening is established with respect to the Cauchy–Schwarz inequality. General properties of the Moore–Penrose inverse of a skew-symmetric matrix are discussed as well.

Keywords

• Cauchy–Schwarz inequality
• Moore–Penrose inverse
• Scalar product
• Cross-product matrix
• Antisymmetric matrix
• Rotation matrix