Geometric measures of uniaxial solids of revolution in higher-dimensional Euclidean spaces and their relation to the second virial coefficient

We provide analytical expressions for the second virial coefficients of various hard, convex, monoaxial solids of revolution in higher-dimensional Euclidean spaces. Therefore, the rotation-averaged mutual excluded volume per particle is calculated employing the Brunn-Minkowski theorem using quermassintegrals of the respective shape. In addition to geometries without singularities in their surface curvature, so far unknown quermassintegrals for geometries with singularities in their surface curvature, such as hyperlenses and hypercones, are calculated. Studying the influence of the detailed particle shape, the second virial coefficients are analyzed in four dimensions depending on the aspect ratio ν. These analytical expressions are extended to arbitrary-dimensional Euclidean spaces. The resulting virial coefficients are compared to available data for analogs in two and three dimensions. For hard hyperspheroids, the universal parity B_{2}^{*}(ν)=B_{2}^{*}(ν^{-1}) of the reduced second virial coefficient with respect to the aspect ratio ν is proven. Unlike other geometric shapes, the excluded volume of hyperspherocylinders in any dimension solely depends on at most three quermassintegrals.