An inverse theorem on ‘economic’ maps

We prove that the bound from the theorem on ‘economic’ maps is best possible. Namely, for $m>n+d$ we construct a map from an $n$-dimensional simplex to an $m$-dimensional Euclidean space for which (and for any close map) there exists a $d$-dimensional plane whose preimage has cardinality not less than the upper bound $\lceil(dn+n+1)/(m-n-d)\rceil+d$ from the theorem on ‘economic’ maps.
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