Section I of this paper provides a necessary and sufficient condition for the existence and uniqueness of the solution of a particular isoperimetric problem, the minimal arc-length problem defined on the set of Quasi-Probability Density Functions (QPDF). This settles the question of existence or non-existence of solution to the minimal arc-length among the eligible Probability Density Functions (PDF). This is accomplished by using the answer to an ancient isoperimetric problem of Queen Dido. Section II provides an alternative proof of the same conclusion by a direct application of an extremal theorem from the Calculus of Variations.

calculus of variation, minimal arc-length problem, class of QPDFs, boundary conditions, legend of Queen Dido, Mathematica.