Rachel Swarns used DNA testing and historical data of slave owners to find out that Michelle Obama has white relatives. Her great grandfather was half white, born to a slave mother and slave owner.
She has this all listed in her book "American Tapestry: The story of the black, white, and multiracial ancestors of Michelle Obama."
This would mean that Michelle Obama is at least 1/32 white. Elizabeth Warren claimed herself to be a minority despite saying she is 1/32 Cherokee.
Here is a picture of Michelle Obama as a baby with her parents and brother.