0

Let $B$ be an $n \times n$ positive-definite matrix and let $r>0$. Consider the hyper-ellipsoid $E_B \subseteq \mathbb R^n$ defined by

$$ E_B(r) := \left\{x \in \mathbb R^n \mid x^\top B x = r \right\}. $$

The hyper-sphere of radius $r$ corresponds to $E_{I_n}(r)$, and it is known to have mean curvature $1/r^2$.

Question. What is the mean curvature of $E_B(r)$, for general $B$ ? Can the mean curvature be written as a function of only $r$ and the condition number of $B$, that is the ratio of the longest to the shortes axes of $E_B(r)$ or equivalently, the ratio of the largest to the smallest singular-values of $B$ ?

dohmatob
  • 9,535
  • I wasn't sure what notion of "curvature" would be appropriate here, that's why I inserted the quotation signs. I will settle for global min and max values of mean curvature. Question update. – dohmatob Dec 29 '21 at 23:17
  • Related – Rodrigo de Azevedo Dec 30 '21 at 07:38
  • 1
    You make it sound like the mean curvature is a constant. This is true only in the case of a sphere. It is the average of the principal curvatures and will vary in a complicated way. So I don't understand what your question can even begin to mean. – Ted Shifrin Dec 30 '21 at 19:04

0 Answers0