Given symmetric positive definite $m\times m$ matrix $\Sigma$, $1 \le p,$ and constant $c > 0$, consider the solution to the optimization problem: $$ \min_{v : \left\|v\right\|_p \ge c} v^{\top}\Sigma v. $$ Using Lagrange's method imposes the restriction on the solution: $$ \Sigma v = k \, |v|^{p-1} \odot \operatorname{sign}\left(v\right), $$ where that's a Hadamard multiply and a Hadamard power on the RHS. When $p=2$, we get $\Sigma v = k v,$ i.e. $v$ is an eigenvector of $\Sigma$ (further consideration of the problem shows it is the eigenvector associated with the smallest eigenvalue, of course.).
When $p \ne 2$, this seems like an odd generalization of 'Eigenvector'. Is this a known generalization? Does this make sense even? Can I reduce the problem back to an eigenvalue problem through a transform somehow? Are there nice forms for $p=1$ or $p=\infty$?