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Portion of planets from which others are invisible

Portion of planets from which others are invisible

Q: Consider a system of n identical stationary spherical planets. For each planet, find all points such that none of the others is visible, and paint them red. How much paint do you need?

Equivalently: shrink-wrap a finite set of identical spheres (ie take their convex hull). What is the area of contact between the spheres and the wrapping?

A: Enough to paint one planet completely. Indeed, if you take all the red portions they can be fitted together without rotation to form one planet (up to boundaries of measure 0).

To see this, consider a ray to infinity, and bring a plane normal to this ray in from infinity until it hits something. If it hits a single planet, the point of contact will be painted red. This establishes a 1-1 correspondence between rays and red points, except that rays whose planes hit multiple planets have no red point. However, the points of contact of such planes form the boundaries of the red regions - such regions are the intersection of finitely many hemispheres, so have piecewise-great-circle boundaries, which are of measure 0.

This argument apparently goes through in all dimensionalities.

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