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which leads to definitions of **weak relaxed witness complex** (or just relaxed witness complex for short) and **strong relaxed witness complex** respectively.
- \image html "swit.svg" "Strong witness witnesses the whole simplex in the witnessing ball"
+ \image html "swit.svg" "Strongly witnessed simplex"
In particular case of 0-relaxation, weak complex corresponds to **witness complex** introduced in \cite de2004topological, whereas 0-relaxed strong witness complex consists of just vertices and is not very interesting.
Hence for small relaxation weak version is preferable.