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In the Onsager model of one-component hard-particle systems, the entire phasebehaviour is dictated by a function of relative orientation, which representsthe amount of space excluded to one particle by another at this relativeorientation. We term this function the excluded volume function. Within thecontext of two-dimensional convex bodies, we investigate this excluded volumefunction for one-component systems addressing two related questions. Firstly,given a body can we find the excluded volume function?, Secondly, can wereconstruct a body from its excluded volume function? The former is readilyanswered via an explicit Fourier series representation, in terms of the supportfunction. However we show the latter question is ill-posed in the sense thatsolutions are not unique for a large class of bodies. This degeneracy is wellcharacterised however, with two bodies admitting the same excluded volumefunction if and only if the Fourier coefficients of their support functionsdiffer only in phase. Despite the non-uniqueness issue, we then propose andanalyse a method for reconstructing a convex body given its excluded volumefunction, by means of a discretisation procedure where convex bodies areapproximated by zonotopes with a fixed number of sides. It is shown that thealgorithm will always asymptotically produce a best $L^2$ approximation of thetrial function, within the space of excluded volume functions of centrallysymmetric bodies. In particular, if a solution exists, it can be found. Resultsfrom a numerical implementation are presented, showing that with only desktopcomputing power, good approximations to solutions can be readily found.

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