Explicit excluded volume of cylindrically symmetric convex bodies

TL;DR

This paper provides an explicit representation of the excluded volume for cylindrically symmetric convex bodies, revealing the absence of a dipolar component and refining existing analytic estimates for spheroids.

Contribution

It introduces a new method to explicitly compute excluded volume for symmetric convex bodies, correcting previous assumptions about shape dipoles and improving analytic estimates.

Findings

Excluded volume lacks a dipolar component for these bodies.

Method validated on cones and spheroids.

Refines classic analytic estimates for spheroids.

Abstract

We represent explicitly the excluded volume Ve{B1,B2} of two generic cylindrically symmetric, convex rigid bodies, B1 and B2, in terms of a family of shape functionals evaluated separately on B1 and B2. We show that Ve{B1,B2} fails systematically to feature a dipolar component, thus making illusory the assignment of any shape dipole to a tapered body in this class. The method proposed here is applied to cones and validated by a shape-reconstruction algorithm. It is further applied to spheroids (ellipsoids of revolution), for which it shows how some analytic estimates already regarded as classics should indeed be emended.