The front corner of both boxes is in fair territory. This part is on the infield side of the foul line, which runs from the outside of 1st/3rd base to the rear corner of home plate. A batted ball first touched by a fielder there or which comes to rest there is a fair ball.
OTOH, it is still the batter's box, so a batted ball that contacts the batter while she is still in the batter's box is a foul ball.
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Tom
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