Petty's equilateral-set conjecture in dimension five

Petty conjectured that every n-dimensional real normed space contains n+1 equidistant points. The conjecture was proved in dimensions three and four by Petty and Makeev, respectively. We prove the five-dimensional case: every five-dimensional real normed space contains six equidistant points. The proof uses a bordism of equilateral configurations and a homological obstruction.