The authors’ goal is to construct some sharp isoperimeter inequalities for the left side of the inequality \[ d(x,z)d(y,t)+ d(x,t)d(y,z)- d(x,y) d(z,t)\geq 0\quad\text{for all} x,\;y,\;z,\;t \] in a metric space \(X\). In Section 2, the authors proved that the measure of non-Ptolemaity is maximal possible for “equilateral” pseudolinear quadruples. In Section 3, external problems on the class of finite pseudometric spaces were discussed. The authors also noted that this class constitutes a natural “minimal extension” of the class of finite metric spaces which guarantees the existence of an extremal for the problems with continuous target functions. Finally, in Section 4, the authors described maximally Ptolemaic pseudometric spaces.