This gives some control over the behaviour of ......
Condition (c) is intended to give us firm control over ......
This extension retains control on ...... at the sacrifice of losing some control on ...... [Not: “loosing”]
The requirement that competitors are symmetric will only be used to get some very rudimentary control on the asymptotics of $G$.
The obstacle we have is that we must control the area of $G$.
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