On a problem of combinatorial geometry

What is it about?

Let $H\coloneq (h_1,h_2,\cdots,h_n)$ be a collection of $n$ hyperplanes in ${\bf R}^d$ $(d\geq 2,\;n>d)$ and let $a_i$ be the defining normal vector of $h_i$, $i=1,\cdots,n$. In the paper a short elementary proof of the following theorem is presented: If any $d$ members of the set $\{a_1,\cdots,a_n\}$ are linearly independent and $\bigcap^n_{i=1}h_i=\emptyset$, then among "monolithic'' members of the decomposition of ${\bf R}^d$ defined by $H$ there are at least $n-d$ simplices. The assumption of this theorem taken on $H$ is a little weaker than the assumption: "$h_i$, $i=1,\cdots,n$, are in general position''. The theorem under the latter stronger assumption on $H$ was proved by Grünbaum and Shephard in 1979, but their proof is longer and is not elementary. Reviewed by Béla Uhrin

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