While searching through different materials(textbooks, internet websites, publications), I came across the script about Borsuk's conjecture (staffwww.fullcoll.edu/dclahane/ma/watsontalk.pdf). In this pdf. file(pages 25-30) very simple, but with gaps, proof of Borsuk's conjecture for 2-dimensional case was presented.
- I wonder about alternative, but also simple proofs for this conjecture in 2d.
- I would be thankful for posting alternative proofs or internet websites below my post.
You can find a short proof in 2d case for instance in
D. Gale, On inscribing n-dimensional sets in a regular n-simplex. Proc. Amer. Math. Soc. 4, (1953) 222–225.
and a short proof in the 3d case in
B. Grünbaum, A simple proof of Borsuk's conjecture in three dimensions. Proc. Cambridge Philos. Soc. 53 (1957), 776–778.