to do something that makes people respect you less
He didn't want to lose face in front of his friends, so he refused to admit he was wrong.
彼は友達の前で面目を失いたくなかったので、自分が間違っていたと認めることを拒否しました。
to become less respected by others because of something you did
She was afraid to lose face by admitting she didn't know the answer.
彼女は答えを知らないと認めることで面目を失うことを恐れていました。