Radon theorem (geometry)

Statement

The theorem of Radon , or lemma of Radon , on the units convex S affirms that any $A unit = \ {a_1, \ dowries, a_ {d+2} \}$ container $d+2$ elements of $\ mathbb {R} ^ {D}$ admits a partition in two $A_1 parts, A_2$ whose convex envelopes $\ mathrm {Conv} (A_1)$ and $\ mathrm {Conv} (A_2)$ meet. Such a partition is then called partition of Radon , and a point of intersection of the envelopes is called not Radon (it is not a question a priori of one of the points $a_i$ ).

Let us take the $d=2$ example. In this case the $A$ unit consists of four points. The partition of $A$ can give a whole of three points and a singleton, the first forming a triangle containing the last point. Or then the partition consists of two units made up each of two points, the segments being intersected in a point.

This result was published for the first time by Johann Radon in 1921. It seems there intermediate result in the proof of the Théorème of Helly, which explains the current denomination of lemma .

Proof

It is supposed that

X = \ {a_1, \ dowries, a_ {d+2} \} \ subset \ mathbf {R} ^d

. Let us consider the system:

\sum_{j=1}^{d+2} \lambda_j a_j=0

$\sum_{j=1}^{d+2} \lambda_j=0$

real unknown factors $\ lambda_1, \ dowries, \ lambda_ {d+2}$ : it is equivalent to a linear system of $d+1$ unknown equations to $d+2$ $a_1, a_2, \ dowries, a_ {d+2}$ , since the first equation, if one develops it in a system for each component of the $a_i$ (vectors in $\ mathbf {R} ^d$ ), is transformed at once into $d$ traditional linear equations. There thus exists a nonnull solution of this system. Let us fix $\ lambda_1, \ lambda_2, \ dowries, \ lambda_ {d+2}$ such a solution. Let us pose then:

I_1 = \ {I \, \ mid \, \ lambda_i > 0 \}

$I_2 = \ {I \, \ mid \, \ lambda_i \ Leq 0 \}$

Since the sum of the $\ lambda_i$ is null whereas the $\ lambda_i$ are not all null, $I$ and $J$ are not empty.

The necessary partition of $A$ is then $A_1= \ {x_i \, \ mid \, I \ in I_1 \}$ and $A_2= \ {x_i \, \ mid \, I \ in I_2 \}$ . Indeed, it is immediate to check starting from the system, that:

\ frac {\ displaystyle \ sum_ {I \ in I_1} a_i x_i} {\ displaystyle \ sum_ {I \ in I_1} a_i} = \ frac {\ displaystyle \ sum_ {I \ in I_2} a_i x_i} {\ displaystyle \ sum_ {I \ in I_2} a_i}

and this formula provides a common point to the convex envelopes of $A_1$ and $A_2$ .

Theorem of Tverberg

Helge Tverberg showed in 1966 a generalization of this theorem for partitions of $A$ in R subsets. The theorem of Tverberg affirms that:

a $A$ unit of $1 + (d+1) (r-1)$ points of $\ mathbb {R} ^d$ admits a partition in $r$ subsets whose intersection of the convex envelopes is not empty

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