# Generalization of Beck's theorem

Beck's theorem is a classical result in discrete geometry which describes about the geometry of points in the plane. The result states that a finite collections of points in the plane fall into one of two extremes; one where a large fraction of points lie on a single line, and one where a large number of lines are needed to connect all the points.

Formally, the statement of the theorem is as follows:

Theorem: Let $S$ be a set of $n$ points in the plane. If at most $(n-k)$ points lie on any line for some $0 \leq k < n-2$, then there exist $\Omega(nk)$ lines determined by the points of $S$.

Now, my question is that if the generalization/variation of the above result is known in the case of higher dimension $(dim\geq 3)$?

• Except few sentences all other sentences of the question are exact copy of wiki. Provide a quotation if you copy that amount of text from somewhere else (not from your own). Finally why you didn't see this? "Take for example a set of 2n points in R3 all lying on two skew lines. Assume that these two lines are each incident to n points. Such a configuration of points spans only 2n planes. Thus, a trivial extension to the hypothesis for point sets in Rd is not sufficient to obtain the desired result.", that says generalization first needs a good new definition rather than the trivial ones. Dec 24 '14 at 12:23

Theorem: For any $$d \geq 2$$, there are constants $$\beta_d, \gamma_d \in (0,1/2]$$ such that for any set $$P$$ of $$n$$ points in $$\mathbb{R}^d$$, at least one of the following holds:
• a hyperplane contains at least $$\beta_d n$$ points of $$P$$
• the $$d$$-tuples of $$P$$ span at least $$\gamma_d n^d$$ distinct hyperplanes.
Here, a hyperplane is a $$(d-1)$$-dimensional flat. For $$d=2$$, this is essentially what you stated. The restriction that $$\beta_d \leq 1/2$$ is from the example of having two skew lines in $$\mathbb{R}^3$$ containing $$n/2$$ points each, where the number of spanned planes is $$n$$. For more, see: http://www.cs.elte.hu/~elekes/Abstracts/elest.ps