Noun
toe box (plural toe boxes)
The portion of the upper of footwear that encloses the toes.
In 1968, Puma SE introduced the first pair of sneakers with Velcro straps in lieu of shoelaces, and these became popular by the 1980s, especially among children and the elderly. citation citation The toe box is the part that covers and protects the toes. Source: Internet
These have a toe box that is stiffened with glue and a hardened sole so the dancer can stand on the tips of their toes. Source: Internet