  The classical Hlawka inequality possesses deep connections with zonotopes and
zonoids in convex geometry, and has been related to Minkowski space. We
introduce Hlawka Type-1 and Type-2 quantities, and establish a Hlawka-type
relation between them, which connects a vast number of strikingly different
variants of the Hlawka inequalities, such as Serre's reverse Hlawka inequality
in the future cone of the Minkowski space, the Hlawka inequality for
subadditive function on abelian group by Ressel, and the integral analogs by
Takahasi et al. Besides, we announce several enhanced results, such as the
Hlawka inequality for the power of measure function. Particularly, we give a
complete study of the Hlawka inequality for quadratic form which relates to a
work of Serre.
