>>12484516This is a pretty easy problem.
Use the triangle inequality to prove that the shortest distance between any point outside the sphere is a straight line towards the sphere’s center.
Suppose the shortest distance between the cone and the sphere is in the interior of the cone. Then the straight line connecting the point to the center of the sphere must pass through the surface of the cone. Obviously, the smallest subsegment of the line (that still connects the cone and sphere center) is one that terminates at the surface of the cone, so any minimal distance point must be on the surface of the cone.
Next, draw any line from the sphere’s center, to the rotation axis of the cone. Take a sphere using the intersection point of the constructed line and cone axis, using the portion of the line segment inside the cone as the radius. By the same triangle inequality logic, any point on the cone’s surface outside the plane containing the cone’s axis, and sphere’s center, is farther away from the sphere than a point on (the closer half of) the intersection of said plane and cone’s surface.
Hence, the line , where .
The rest is left as an exercise.
