Recent extensive studies reveal that surfactant-stabilized spherical alkane emulsion droplets spontaneously adopt polyhedral shapes upon cooling below a temperature T-d while remaining liquid. Further cooling induces the growth of tails and spontaneous droplet splitting. Two mechanisms were offered to account for these intriguing effects. One assigns the effects to the formation of an intradroplet frame of tubules consisting of crystalline rotator phases with cylindrically curved lattice planes. The second assigns the sphere-to-polyhedron transition to the buckling of defects in a crystalline interfacial monolayer, known to form in these systems at some T-s > T-d. The buckling reduces the extensional energy of the crystalline monolayer's defects, unavoidably formed when wrapping a spherical droplet by a hexagonally packed interfacial monolayer. The tail growth, shape changes, and droplet splitting were assigned to the decrease and vanishing of surface tension, gamma. Here we present temperature-dependent gamma(T), optical microscopy measurements, and interfacial entropy determinations for several alkane/surfactant combinations. We demonstrate the advantages and accuracy of the in situ gamma(T) measurements made simultaneously with the microscopy measurements on the same droplet. The in situ and coinciding ex situ Wilhelmy plate gamma(T) measurements confirm the low interfacial tension, less than or similar to 0.1 mN/m, observed at T-d. Our results provide strong quantitative support validating the crystalline monolayer buckling mechanism.