Referring to the sketch above as option 1

you can have two option, option 1, as obvious cutout at the middle of the u shape staircase, in addition to this you can also have a double height entrance foyer with a chandelier in between hanging from the top. on first floor you can shift the balcony door side ways as mentioned with a circle.

option 2 would be a complete change of staircase orientation , bringing the staircase in front foyer, extending the upper cutout as marked in dotted line, the new found area can help you extend your both bathrooms areas or can help you put some ground floor furniture or a informal study area.(see if your FSI permits to all these changes). and on the top your new area can accommodate your big gym equipment's like treadmill , and some additional gym storage. here you can also get to know from top who has arrived to your front door peeking downwards.

further the walls of this area can be decorated while doing interiors and make the foyer look grand & luxurious.

option three,

If you are thinking of giving a cutout in center of the hall, please avoid, your areas above wont be usable in an appropriate way.

cheers