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Convex polytopes are the convex hulls of a finite set of points in Euclidean spaces. They have rich combinatorial, arithmetic, and metrical theory, and are related to toric varieties and to linear programming

18 votes
Accepted

On the number of Archimedean solids

A proof of the enumeration theorem for the Archimedean solids (which basically dates back to Kepler) can be found in the beautiful book "Polyhedra" by P.R. Cromwell (Cambridge University Press 1997, p …
Andrey Rekalo's user avatar
7 votes

Area of cross-section (at midpoint perpendicular to longest diagonal) in the unit cube of di...

This is a very old problem and there is a classical analytic approach to it. You can express the volume of sections of a convex body in terms of the Fourier transform of powers of the Minkowski functi …
Andrey Rekalo's user avatar
4 votes
Accepted

A variation on "Hearing the shape of a drum" for polytopes.

The short answer is that there are no particular constraints on the spectral decomposition of the function $\varphi$, as long as a basic convexity condition is satisfied. Lemma..Assume that $\var …
Andrey Rekalo's user avatar