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M. Goresky and R. MacPherson intersection homology is also defined from thesingular chain complex of a filtered space by H. King, with a key formula tomake selections among singular simplexes. This formula needs a notion ofdimension for subspaces $S$ of a Euclidean simplex, which is usually taken asthe smallest dimension of the skeleta containing~$S$. Later, P. Gajer employedanother dimension based on the dimension of polyhedra containing $S$. This lastone allows traces of pullbacks of singular strata in the interior of the domainof a singular simplex.  In this work, we prove that the two corresponding intersection homologies areisomorphic for Siebenmann's CS sets. In terms of King's paper, this means thatpolyhedral dimension is a ``reasonable'' dimension. The proof uses aMayer-Vietoris argument which needs an adaptated subdivision. With thepolyhedral dimension, that is a subtle issue. General position arguments arenot sufficient and we introduce strong general position. With it, a stabilityis added to the generic character and we can do an inductive cutting of eachsingular simplex. This decomposition is realised with pseudo-barycentricsubdivisions where the new vertices are not barycentres but close points ofthem.

A reasonable notion of dimension for singular intersection homology