An ambit is the area or range that someone controls or affects. The kid who's voted president of her sixth grade class is going to be thrilled, despite her relatively small ambit.
The origins of ambit go back to the Latin word for "going about or going round," ambitus, and for a time, ambit literally meant "the area surrounding a building." Today, the range of a person or group's power is their ambit. For example, if a crime is committed on a college campus, the town police won't investigate if they decide that it's the ambit of the college police, or under their jurisdiction.