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A solid ellipsoid is the filled region bounded by an ellipsoid. The open version centered at the origin iswhere $Q$ is a real symmetric positive-definite matrix; replacing $<$ by $\le$ includes the boundary. Its semi-axis lengths are ${\lambda }_i^{-1/2}$, where ${\lambda }_i$ are the eigenvalues of $Q$. In three dimensions its volume is $4\pi /(3\sqrt{\det Q})$, by a change of variables from the unit ball.

In even-dimensional symplectic geometry, a uniform dilation $x\mapsto tx$, $t>0$, multiplies the symplectic capacity by $t^2$. This concerns one common factor in every coordinate. Independent changes in the different semi-axis lengths are not a uniform dilation and have no such common square factor.