Some preliminiaries: since the recent HTML-protocoll doesn't allow the use of the ordinary mathematical symbols, some substitutes: P denotes the power set of a set, that means the set of all subsets. A denotes "for All", E denotes "there Exists a(n)". McN means that M is a subset of N (not necessairily a real one), xeM means that x is a member of M, <= means less or equal, and log means the logarithm with base 2. R is the set of the real numbers.

We regard subsets of Rⁿ, with fixed n. The mapping p operates on this subsets.

Definition: p: P(Rⁿ)->P(Rⁿ)
p(M):={z|E x e M, E y e M, E r e [0,1]: z = rx + (1-r)y}

Obviously M c p(M). Also it is true: M convex <=> M = p(M), since this is an immediate consequence of the definition of convexity.
If we apply the mapping p several times on M, we would expect the result sometime to become convex, so that further applications will result in no change. The smallest number k with p^k(M) = p^k+1(M) we will call the "concavity" of M. A set is convex iff its concavity is zero. We will see that such a number always exist.
An interesting question is what the smallest number k is that with given number n all M c Rⁿ p^k(M) = p^k+1(M), that means, the question what the greatest possible amount of concavity for a subset of Rⁿ is.
Investigating this, we can restrict ourself to sets M consisting of nothing but (n+1) single points, with n of those points being linear independent, that means, we consider one of those points as origin, while the remaining n points form a coordinate system.
Why can we restrict our investigation that way?

Sets containing less elements are of no interest, since they can be regarded as subset of R^n-1. But a set of the described nature will sometime, after k applications of p, form a n-simplex. Why are additional points of no importance? Well, we are interested for points z being in p^k, but not in p^k-1. Than there is a n-simplex containing z with its vertices being elements of M. Why? Such a point as z results from not more than 2^k points of M, that means, it is inside a n-dimensional polyhedron with its vertices formed by 2^k ponts of M. But this n-polyhedron can in any cases be divided into n-simplices with there vertices being vertices of the n-polyhedron (this must be proven, but it is a usual standard theorem of topology or geometry). So z is not only inside a n-polyhedron with 2^k vertices, but also inside a n-simplex with (n+1) vertices being elements of M. Also all we need to regard, if we are interested in z, are those (n+1) vertices. Taking more elements can cause nothing but that z can be constructed earlier, that means, more elements can make k only smaller.

If we apply p to M, we obtain in the first step lines, that means objects with dimension 1. With the next step, applying p again, the lines are connected with lines, resulting in a space. In every step, two (linear independend) manifolds of dimension d are connected with lines, so the dimension of the result is (2d+1): one d is distributed by every manifold, the additional dimension is distributed by the connection (the [0,1] of the definition of p). Of course all this works only if the underlaying space allows as much dimensions. That means, the set we start with has dimension d=0, every application of p increases the dimension d to max(2d+1, n), and that means, after k applications of p we get the dimension max(2^k-1, n). p becomes stationary, if p^k(M) is a n-simplex, that means, the dimensionality stops changing. So if we are searching the smallest k such that p^k is stationary, we have to search the smallest k such that n <= 2^k -1, or, transformed, the smallest integer k, such that log(n+1) <= k.

With n=2 and n=3, both times k=2, that means: on a plane, and also in space, p²(M) is convex for every M.

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