This question is the title of a lecture by Craig B. Thompson, President and CEO of Memorial Sloan-Kettering Cancer. He's talking about the latest findings in cancer research.
The answer focusses on the state of our genes and specifically those responsible for glucose metabolism. A PET scan checks this.
Glucose is the trigger for dormant cancer cells to proliferate.
'It matters where your calories come from. We have good evidence that excess fat does not increase your cancer risk, excess carbohydrate does increase your cancer risk and protein is somewhere in the middle.'
As Tom Naughton comments, 1/3 of the student audience had misunderstood the lecture and still believed that dietary fat would increase the growth of tumours. Low fat health messages have taken hold on public perception.