Octupolar approximation for the excluded volume of axially symmetric convex bodies

We propose a simply computable formula for the excluded volume of convex, axially symmetric bodies, based on the classical Brunn-Minkoski theory for convex bodies, which is briefly outlined in an Appendix written in a modern mathematical language. This formula is applied to cones and spherocones, which are regularized cones; a shape-reconstruction algorithm is able to generate the region in space inaccessible to them and to compute their excluded volume, which is found to be in good agreement with our approximate analytical formula. Finally, for spherocones with an appropriately tuned amplitude, we predict the occurrence of a relative deep minimum of the excluded volume in a configuration lying between the parallel alignment (where the excluded volume is maximum) and the antiparallel alignment (where the excluded volume is minimum).