Some inequalities for tetrahedra (Q2042133)

For a tetrahedron \(T\), the intrinsic and extrinsic radii are defined by \[ \operatorname{Rad}(T):=\min_{x\in T}\max_{y\in T}\rho(x,y), \quad \operatorname{rad}(T):=\min_{x\in T}\max_{y\in T}\|x-y\|, \] respectively, where \(\rho(x,y)\) is the geodesic length between \(x\) and \(y\). Similarly, the intrinsic and extrinsic diameters of the tetrahedron are \[ \operatorname{Diam}(T):=\max_{x, y\in T}\rho(x,y), \quad \operatorname{diam}(T):=\max_{x, y\in T}\|x-y\|. \] The authors prove double-sided inequalities for the ratios of all six combinations of these quantities. Some of them are not sharp, as for example \[ 1<\frac{\operatorname{Diam}(T)}{\operatorname{Rad}(T)}\le2, \] which the authors conjecture that can be improved to \(\frac {2}{\sqrt{3}}\le\frac{\operatorname{Diam}(T)}{\operatorname{Rad}(T)}\).