Return to Revisions

Let $O\in \partial K$. Call $P\in \partial K$ a critical point if $\langle O-P, X-P\rangle \geq 0$ for all $X\in K$. Note that if $P_1$ and $P_2$ are critical points with $\angle P_1 O P_2 \leq \frac{\pi}{3}$ then by the Pythagorean theorem $\frac{|OP_1|}{|OP_2|}\leq 2$. Therefore by a simple packing argument $\inf_{P \; critical} |OP|>0$. For every $C>0$ smaller than this infimum, we show that the sphere $S_C$ of radius $C$ centered at $O$ has the property that $S_C\cap K$ is connected. Indeed, if points $X,Y$ were in different connected components of $S_C\cap K$, we would connect them by a path in $K$ and then "push out" the path away from $O$ using a flow in $K$, using that fact that the flow can only get stuck at a saddle point of $\partial $K$ for the distance function, and saddle point is necessarily a critical point. The construction of the flow in the absence of such Grove-Shiohama critical points was described in http://link.springer.com/article/10.1007/BF02187719