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If an obese person and a "fit" person were marooned on an island where there was plenty of drinkable water but literally nothing to eat; all else being equal, would the obese person stay alive for significantly longer than the "fit" person?

(Let's assume that the weather remains mild and tolerable, and there is adequate shelter and opportunity for restful sleep)

Related framing of the question: Could a person intentionally starve themselves by depending solely on their body fat as a substitute for food intake (perhaps also taking some essential multivitamins, if that is necessary)?

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Assuming no underlying health problems and similar muscle stores, the obese person would survive significantly longer.

When eating normally, the body gets most of its energy from free glucose and glycogen stores. After a day or two of no food intake, the body will switch to get most of its energy from the breakdown of fat tissue. Some cell types can't utilize energy from fat, so those cells will get energy from the breakdown of muscle protein which can be converted to glucose (Biochemistry Textbook - 30.3.1).

Death from starvation can occur due to running out of fat tissue, or running out of muscle tissue that can be safely broken down. A person of average body weight would generally run out of fat tissue first, whereas an obese person would run out of muscle tissue first (Review Article - see pg. 15). Assuming they started with the same amount of muscle tissue, that means the obese person would survive longer.

A caveat: The immune system is significantly weakened in starvation due to metabolic changes, making deadly infections common (WHO). An obese person would not have advantage against those infections.

To answer your second framing:

The average person can survive a long time without eating - about 1 to 3 months (Biochemistry Textbook - 30.3.1) (NOTE: This does NOT mean a person can safely fast for more than a day or so at a time. After a few days there will be significant, often permanent damage to many organs). That said, vitamin deficiencies take even longer to develop in most cases (Victoria Dept. of Health), so taking a multivitamin wouldn't help much.

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Yes ... unless the fit person uses their advantage to kill and eat the other.

This is probably the evolutionary cause of obesity: if you have an irregular food supply, it is sensible to build up a stock, either externally stored food or internally stored fat. This gave us physiology, and preferences, that work less well in rich countries, where you are surrounded 24/7 by a buffet.

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