The inradius \(r(K)\), the circumradius \(R(K)\), the diameter \(D(K)\) and the minimal width \(\Delta(K)\) are classical fundamental functionals of a convex body \(K.\) Some of these intermediate functionals are well-known functionals in approximation theory, called Bernstein and Kolmogorov diameters, and they are also of interest for computational aspects of convex bodies. The authors study the following generalization of the inradius: For a convex body \(K\) in the \(d-\)dimensional Euclidean space \(E^d\) and a linear \(k-\)plane \(L\) the inradius of \(K\) with respect to \(L\) is defined by \(r_L(K)=\max\{r(K;x+L):x\in E^d\},\) where \(r(K;x+L)\) denotes the ordinary inradius of \(K\cap(x+L)\) with respect to the affine plane \(x+L.\) The generalized inradius \(r_L(P)\) is determined for polytopes, and \(\min\{r_L(T_r^d)\): \(L\) is a \(k-\)plane\(\}\) is estimated for the regular \(d-\)simplex \(T_r^d.\)