I'm not sure whether one factor in a resume dominates all others. If your GPA is slightly below the school's range, then it's still possible to compensate for this in other areas. A very strong GMAT score, a degree from a highly regarded school, or excellent work experience can at least partly make up for this deficiency. It also depends on the program which you apply to; admissions for the MSc Finance, for example, is much more strict than than admissions for the MSc Economic History.
With that said, LSE is one of the premier universities in the world. Many promising people with excellent backgrounds apply every year, and they represent your competition. Another issue to consider is that a master's degree focuses on a specific field. Consequently, admissions expects a developed and specific skill set not necessarily found in an MBA. If this isn't demonstrated in the application, then one's chances become limited.
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