Category Archives: Pretty little things

‘Shear’ convex polyhedra: solution

Here I describe the solution to the problem from the last post. Spoilers! Advertisements

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‘Shear’ convex polyhedra

I came up with what seems quite a beautiful geometric construction solving the problem described below. On a compact convex set    consider the function where we denote  In other words, “hanging” in the point    we measure the “width” of the … Continue reading

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How closed form conjectures are made

Discovered an interesting discussion on math.stackoverflow.

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Arrow’s impossibility theorem

There is an amazing result in the theory of social choice called Arrow’s impossibility theorem. Suppose we have a set    of    possible alternatives, say, election candidates, and    voters. Each voter organizes its individual preference list over    that is, a linear ordering of    (suppose … Continue reading

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Cutting hypercube by an affine subspace

Fix an integer    Cut hypercube    with an arbitrary  -dimensional affine subspace    of general position.  How many vertices    has the obtained polyhedron? It occurs that    grows polynomially, and not exponentially, with    The goal of this … Continue reading

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Mutual сoherence III

It turned out that the proof of Welch bound for mutual coherence, which I previously mentioned here, is very simple. Apparently it can also be obtained from Gershgorin circle theorem.

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Stereographic projection

Found a nice Greek-style purely geometrical proof of the main property of stereographic projection, the fact that it preserves circles.

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