Olivia Judson compiles the evidence:
[O]ne long-term study of more than 6,500 people in northern California found that those who were fat around the middle at age 40 were more likely to succumb to dementia in their 70s. A long-term study in Sweden found that, compared to thinner people, those who were overweight in their 40s experienced a more rapid, and more pronounced, decline in brain function over the next several decades.
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