The NYT’s Well blog has the details. The brain’s hippocampus is a critical center for memory. Alzheimers disease is associated with a gene called apo-E4. Carriers of that gene who exercise regularly have less shrinkage of the hippocampus than non-exercisers.
To PROVE that regular exercise prevents dementia-related shrinkage of the hippocampus, you’d have to force some folks to exercise and stop others who wanted to exercise. A couple years later, scan their brains and compare the two groups. That study may never be done.
The Mediterranean diet also seems to prevent or forestall dementia.