Radii of simplices and some applications to geometric inequalities (Q1878963)

In any Euclidean space \({\mathbf E}^d\), bodies have circumradii and inradii (though not necessarily unique inspheres). Moreover, for dimensions \(1\leq j\leq d\), we can define the outer \(j\)-radius \(R_j(B)\) of a body \(B\) as the smallest radius such that some \(j\)-ball of that radius contains the projection of \(B\) into some \(j\)-dimensional subspace, and the inner \(j\)-radius \(r_j(B)\) as the radius of the largest \(j\)-ball contained in \(B\). This paper is largely a (very readable) review of the literature on inequalities between these various radii. Some new results are given, notable a counterexample showing that a certain theorem of Gritzmann and Klee about circumradii does not generalize to outer \(j\)-radii with \(j<d\).