  We provide analytical expressions for the second virial coefficients of hard,
convex, monoaxial solids of revolution in ${\mathbb{R}^{4}}$. The excluded
volume per particle and thus the second virial coefficient is calculated using
quermassintegrals and rotationally invariant mixed volumes based on the
Brunn-Minkowski theorem. We derive analytical expressions for the mutual
excluded volume of four-dimensional hard solids of revolution in dependence on
their aspect ratio $\nu$ including the limits of infinitely thin oblate and
infinitely long prolate geometries. Using reduced second virial coefficients
$B_2^{\ast}=B_2/V_{\mathrm{P}}$ as size-independent quantities with
$V_{\mathrm{P}}$ denoting the $D$-dimensional particle volume, the influence of
the particle geometry to the mutual excluded volume is analyzed for various
shapes. Beyond the aspect ratio $\nu$, the detailed particle shape influences
the reduced second virial coefficients $B_2^{\ast}$. We prove that for
$D$-dimensional spherocylinders in arbitrary-dimensional Euclidean spaces
${\mathbb{R}^{D}}$ their excluded volume solely depends on at most three
intrinsic volumes, whereas for different convex geometries $D$ intrinsic
volumes are required. For $D$-dimensional ellipsoids of revolution, the general
parity $B_2^{\ast}(\nu)=B_2^{\ast}(\nu^{-1})$ is proven.
