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Shape functionals play a large role in many physical applications [40,41,42], and have been subject to extensive mathematical investigation [40]. In the present paper, these functionals of particle shape are normalized so as to be independent of the size of the particle and to equal unity for a sphere. The shape functionals considered in this paper include: the surface area, the mean radius of curvature, the radius of gyration, the electrostatic capacity, the excluded volume (binary cluster integral for purely repulsive particles in the theory of non-ideal gases), and the intrinsic conductivity for both insulating and superconducting objects in a normal conducting matrix. For ellipsoids of revolution, these are all given by simple analytic formulas. To make these shape functionals equal to unity for a sphere and independent of absolute particle size, we normalize them by the same property defined for a sphere with equal volume to the ellipsoid, i.e. Vsph = (4π / 3) r3 = (4π / 3) ab2, where a is the length of the symmetry axis, b is the length of each axis perpendicular to the symmetry axis, and a/b is the aspect ratio of the ellipsoid.
A striking qualitative feature of the shape functionals we investigate are the general "isoperimetric relations" [40] that show that these functionals tend to be minimized for objects having more symmetric shapes, with absolute minima existing for the sphere, the most symmetric object having a finite volume. The explicit illustration of these famous isoperimetric inequalities for ellipsoidal particles and for a variety of properties should have independent interest for the insight it provides into the interrelation between important particle properties.