The excluded volume of two-dimensional convex bodies: Shape reconstruction and non-uniqueness

TL;DR

This paper explores the relationship between convex body shapes and their excluded volume functions in 2D, revealing non-uniqueness in reconstruction and proposing an approximation method using zonotopes.

Contribution

It characterizes the non-uniqueness in reconstructing convex bodies from their excluded volume functions and introduces a practical approximation algorithm.

Findings

Reconstruction from excluded volume functions is generally non-unique.

Fourier coefficients of support functions determine equivalence classes.

The proposed algorithm approximates bodies with zonotopes effectively.

Abstract

In the Onsager model of one-component hard-particle systems, the entire phase behaviour is dictated by a function of relative orientation, which represents the amount of space excluded to one particle by another at this relative orientation. We term this function the excluded volume function. Within the context of two-dimensional convex bodies, we investigate this excluded volume function for one-component systems addressing two related questions. Firstly, given a body can we find the excluded volume function?, Secondly, can we reconstruct a body from its excluded volume function? The former is readily answered via an explicit Fourier series representation, in terms of the support function. However we show the latter question is ill-posed in the sense that solutions are not unique for a large class of bodies. This degeneracy is well characterised however, with two bodies admitting the same excluded volume function if and only if the Fourier coefficients of their support functions differ only in phase. Despite the non-uniqueness issue, we then propose and analyse a method for reconstructing a convex body given its excluded volume function, by means of a discretisation procedure where convex bodies are approximated by zonotopes with a fixed number of sides. It is shown that the algorithm will always asymptotically produce a best $L^2$ approximation of the trial function, within the space of excluded volume functions of centrally symmetric bodies. In particular, if a solution exists, it can be found. Results from a numerical implementation are presented, showing that with only desktop computing power, good approximations to solutions can be readily found.