AbstractLet {n\geq 2} be an integer, {P=\mathrm{diag}(-I_{n-\kappa},I_{\kappa},-I_{n-\kappa},I_{\kappa})} for some integer {\kappa\in[0,n]}, and let {\Sigma\subset{\mathbb{R}}^{2n}} be a partially symmetric compact convex hypersurface, i.e., {x\in\Sigma} implies {Px\in\Sigma}, and {(r,R)}-pinched.
In this paper, we prove that when {{R/r}<\sqrt{5/3}} and {0\leq\kappa\leq[\frac{n-1}{2}]}, there exist at least {E(\frac{n-2\kappa-1}{2})+E(\frac{n-2\kappa-1}{3})} non-hyperbolic P-invariant closed characteristics on Σ.
In addition, when {{R/r}<\sqrt{3/2}}, {[\frac{n+1}{2}]\leq\kappa\leq n} and Σ carries exactly nP-invariant closed characteristics, then there exist at least {2E(\frac{2\kappa-n-1}{4})+E(\frac{n-\kappa-1}{3})} non-hyperbolic P-invariant closed characteristics on Σ,
where the function {E(a)} is defined as {E(a)=\min{\{k\in{\mathbb{Z}}\mid k\geq a\}}} for any {a\in\mathbb{R}}.
On the affine diameter of closed convex hypersurfaces
