A generalized Hlawka's inequality says that for any n (greater than or equa
l to 2) complex numbers x(1), x(2),...,x(n),
Sigma (n)(i=1)\x(i) - Sigma (n)(j=1) x(j)\ less than or equal to Sigma (n)(
i=1) \x(i)\ + (n-2)\ Sigma (n)(j=1)x(j)\.
We generalize this inequality to the trace norm and the trace of an n x n m
atrix A as
parallel toA - TrA parallel to (1) less than or equal to parallel toA paral
lel to (1) + (n-2)\ TrA \.
We consider also the related inequalities for p-norms (1 less than or equal
to p less than or equal to infinity) on matrices.