According to the largest and most precise research, being overweight cuts the risk of dementia.
British researchers admit they were surprised by the findings, which run contrary to current health advice.
The analysis of nearly two million people, in the Lancet Diabetes & Endocrinology, showed underweight people had the highest risk.
Dementia charities still advised not smoking, exercise and a balanced diet.
Dementia is one of the most pressing modern health issues. The number of patients globally is expected to treble to 135 million by 2050.
There is no cure or treatment, and the mainstay of advice has been to reduce risk by maintaining a healthy lifestyle.
The team at Oxon Epidemiology and the London School of Hygiene and Tropical Medicine analyzed medical records from 1,958,191 people aged 55, on average, for up to two decades.
Their most conservative analysis showed underweight people had a 39% greater risk of dementia compared with being a healthy weight.
Those who were overweight had an 18% reduction in dementia – and the figure was 24% for the obese.
Any explanation for the protective effect is distinctly lacking. There are some ideas that vitamin D and E deficiencies contribute to dementia and they may be less common in those eating more.
Heart disease, stroke, diabetes, some cancers and other diseases are all linked to a bigger waistline.