This question was asked on https://math.stackexchange.com/questions/4014421/circular-disk-in-a-convex-domain1 but got no reply
Let $\partial \mathcal D$ be the smooth boundary of a convex domain $\mathcal D$ in the plane and let $\kappa (P)$ denote the curvature of the osculating circle at point $P \in \partial \mathcal D$. Denote the point of maximum curvature by $P^*$ so that $\underset{P}{\text{max}} \{\kappa (P)\} = \kappa (P^*)$ and consider a circular disk $\mathcal C$ whose radius is $r^*=\frac{1}{\kappa (P^*)}$.
Is it true that if this circular disk $\mathcal C(r^*)$ is placed (from inside) tangential to $any$ boundary point then it is completely inside the convex domain $\mathcal D$ and for a similar question with dimension higher than 2?
