$\begingroup$

Is it possible to distinguish a concave polyhedron from a convex one by mesurements made only on its surface, without a reference to the 3d space around it?

$\endgroup$

2 Answers

$\begingroup$

No. Consider (a) a pyramid atop a cube (convex), and (b) a cube with a pyramidal piece cut out of one face (not convex). As intrinsic surfaces, these are isometric.

$\endgroup$ 1 $\begingroup$

In general, any polyhedron, subtending a solid angle $\Omega<2\pi$ sr at each of its vertices, is called a convex polyhedron.

While any polyhedron, subtending a solid angle $\Omega>2\pi$ sr at any of its vertices, is called a concave polyhedron.

It is very practical that a convex polyhedron has its each vertex elevated (protruding outward) while a concave polyhedron has any of its vertices indented on the surface.

$\endgroup$

Your Answer

Sign up or log in

Sign up using Google Sign up using Facebook Sign up using Email and Password Submit

Post as a guest

Post Your Answer Discard

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy