Solid ellipsoid ($E=\{x:x^{T}Qx<1\},Q>0$)

The interior bounded by an ellipsoid is a solid ellipsoid. Positive coordinate dilations multiply its symplectic capacity by the square of the dilation factor.

For the solid ellipsoid $D=\{x^2/a^2+y^2/b^2+z^2/c^2\le 1\}$ with positive semi-axes, set $x=au$, $y=bv$, $z=cw$. The Jacobian determinant of this inverse coordinate map is $abc$, and $D$ becomes the unit ball. Symmetry gives ${\int }_{B_1}{u}^{2}dV=\frac{1}{3}{\int }_{B_1}{r}^{2}dV=4\pi /15$, proving the displayed moment. The other two moments follow by interchanging the semi-axes; mixed moments vanish by reflection symmetry.