# Intersecting hyperplanes.

I am looking for reference for following lemma.

Consider a set of hyperplanes $H$ in a $n$-dimensional space. Let $S$ be the set of intersections of all elements of $H$.
Take a point $w$. Repeat as follows: project $w$ on a hyperplane in $H$. If you repeat this, you will get closer to $S$. If you use all hyperplanes infinitely often, you converge to $S$.

In what books I can find such sort of results?

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## 1 Answer

I've heard this type of scheme called the "method of sequential projections" and the best place to look for references is in the convex optimization literature.

The earliest result of this kind that I'm aware of is by Cheney and Goldstein, who prove the theorem for two convex sets: http://www.ams.org/journals/proc/1959-010-03/S0002-9939-1959-0105008-8/S0002-9939-1959-0105008-8.pdf

Bregman generalizes it to cyclic projections over multiple convex sets: http://www.sciencedirect.com/science/article/pii/0041555367900407#

A more recent survey of the various generalizations e.g. not requiring cyclic projections can be found in Bauschke and Borwein's https://people.ok.ubc.ca/bauschke/Research/05.pdf

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