11.19. Three identical cylindrical surfaces of radius $R$ with mutually perpendicular axes touch each other pairwise.

a) What is the radius of the smallest sphere that touches these cylinders?

b) What is the radius of the largest cylinder that touches the three given ones, the axis of which passes inside the triangle with vertices at the points of contact of the three given cylinders?
11.19. There exists a parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$, the edges $A A_{1}, D C$ and $B_{1} C_{1}$ of which lie on the axes of the given cylinders (problem 1.19); it is clear that this parallelepiped is a cube with edge $2 R$.

a) The center of this cube is at a distance of $\sqrt{2} R$ from all edges, and any other point is at a greater distance than $\sqrt{2} R$ from at least one of the lines $A A_{1}, D C, B_{1} C_{1}$ (problem 1.31). Therefore, the radius of the smallest sphere that touches all three cylinders is $(\sqrt{2}-1) R$.

b) Let $K, L$ and $M$ be the midpoints of the edges $A D, A_{1} B_{1}$ and $C C_{1}$, i.e., the points of pairwise tangency of the given cylinders. Then the triangle $K L M$ is equilateral, and its center $O$ coincides with the center of the cube (problem 1.3). Let $K^{\prime}, L^{\prime}$ and $M^{\prime}$ be the midpoints of the edges $B_{1} C_{1}, D C$ and $A A_{1}$; these points are symmetric to the points $K, L$ and $M$ with respect to $O$. We will prove that the line $l$, passing through the point $O$ perpendicular to the plane $K L M$, is at a distance of $\sqrt{2} R$ from the lines $B_{1} C_{1}, D C$ and $A A_{1}$. Indeed, $K^{\prime} O \perp 6$ and $K^{\prime} O \perp B_{1} C_{1}$, so the distance between the lines $l$ and $B_{1} C_{1}$ is $K^{\prime} O=\sqrt{2} R$; the proof for the other lines is analogous. Therefore, the radius of the cylinder with axis $l$, touching the three given cylinders, is $(\sqrt{2}-1) R$. It remains to check that the distance from any line $l$, intersecting the triangle $K L M$, to one of the points $K^{\prime}, L^{\prime}, M^{\prime}$ does not exceed $\sqrt{2} R$. For example, if the point $X$ of intersection of the line $l^{\prime}$ with the plane $K L M$ lies inside the triangle KOL, then $M^{\prime} X \leqslant \sqrt{2 R}$.