When Heraclius was emperor at the time of the Muslim conquest the Byzantines still ruled Italy and the south of Spain. My supposition is that had the Byzantines held against the initial Arab attack they would've extended their rule across Western Europe (in particular Gaul) long before the Crusades. It was due to the post-Arab decline that Italy was lost by the Byzantines (and of course the Muslims conquered Spain).
I don't view France as a southeastern province.
By the way, have a great Thanksgiving. I am retiring to the desolation of the desert for three days, to ponder the works of Mohammed.
I'm referring to Egypt and Syria as the southeastern provinces. Had the Byzantines not lost Egypt and Syria, they should've been able to hold Italy and Spain. It was the loss of Egypt and Syria that cast Byzantium into two centuries of decline.