From: Dan Asimov <dasimov@earthlink.net> To: math-fun <math-fun@mailman.xmission.com> Sent: Sun, January 2, 2011 11:54:06 PM Subject: [math-fun] Cylinder puzzle Let C_k, k = 1,2,3, . . . , n, . . . be solid unit cylinders in 3-space whose axes all contain the origin. Let X denote the intersection of all the C_k's. Prove that the surface area of X is exactly three times its volume. --Dan _______________________________________________ Here is a proof for finitely many cylinders. Consider a cone of solid angle dΩ with vertex at the origin. Except for a set of measure zero, the cone intersects the boundary of X on a single cylinder. If r is the distance from the origin to the boundary point on the cone axis, then the cone has volume dV = r^3 dΩ/3, and intercepts surface area dS = r^2 dΩ sec i, where the inclination angle i is that between the cone axis and the normal to the surface element. For cylinders of unit radius r = sec i, so dS = 3 dV. -- Gene