Closer look at sum uncertainty relations and related relations

We analyze the weak and critical points of various uncertainty relations that follow from the inequalities for the norms of vectors in the Hilbert space of states of a quantum system. There are studied uncertainty relations for sums of standard deviations, for sums of variances, and other relations between standard deviations or variances. The obtained results are compared with the conclusions obtained in similar cases using the standard Heisenberg-Robertson uncertainty relation. We also show that there exists an upper bound on the product of standard deviations that appears in the Heisenberg-Robertson uncertainty relation.