= Solid ellipsoid
{title2=$E=\{x:x^TQx<1\},\quad Q>0$}
A <solid ellipsoid> is the filled region bounded by an <ellipsoid>. The open version centered at the origin is
$$
E=\{x:x^{\mathsf T}Qx<1\},
$$
where $Q$ is a real <symmetric positive-definite matrix>; replacing $<$ by $\leq$ 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 \b[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.
Back to article page
