Spheres, Platonic solids, Archimedean solids etc are purely convex. This means that if you draw any line connecting two points on the surface of these solids, every point on the line lies inside the solid. We can make "inside" more rigorous by saying the solid divides 3d space into two regions and the finite one is the "inside" of it. Now, imagine a purely concave, finite solid. If you draw any line connecting two points on its surface, every point on the line will always be "outside" the solid. Such a solid can't exist. To make the inside finite, it must "curve around" at some places and those places make it locally convex. How would one go about proving this with Mathematical rigor?
2026-03-27 02:02:57.1774576977
Prove that purely concave finite 3d object can't exist.
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The mathematical word for "finite" here is probably "compact", which means closed and bounded. Let's just look at the surface of such a solid: this is some surface sitting in 3d space. Since the surface is bounded, we could put a giant sphere around the surface, enclosing it but touching it nowhere. We could then slowly shrink the sphere until it first touches the surface. The point where this first touches has to be at least as curved as the sphere, and will be a "convex spot" of the surface.
One of the good mathematical replacements for "convex spot" would be "point with positive Gaussian Curvature".