  We formulate certain inequalities for the geometric quantities characterizing
causal diamonds in curved and Minkowski spacetimes. These inequalities involve
the red-shift factor which, as we show explicitly in the spherically symmetric
case, is monotonic in the radial direction and it takes its maximal value at
the centre. As a byproduct of our discussion we re-derive Bishop's inequality
without assuming the positivity of the spatial Ricci tensor. We then generalize
our considerations to arbitrary, static and not necessarily spherically
symmetric, asymptotically flat spacetimes. In the case of spacetimes with a
horizon our generalization involves the so-called {\it domain of dependence}.
The respective volume, expressed in terms of the duration measured by a distant
observer compared with the volume of the domain in Minkowski spacetime,
exhibits behaviours which differ if $d=4$ or $d>4$. This peculiarity of four
dimensions is due to the logarithmic subleading term in the asymptotic
expansion of the metric near infinity. In terms of the invariant duration
measured by a co-moving observer associated with the diamond we establish an
inequality which is universal for all $d$. We suggest some possible
applications of our results including the comparison theorems for entanglement
entropy, causal set theory and fundamental limits on computation.
