This is part of a famous mosaic showing women exercising and especially playing with balls. The woman on the left is using hand weights. The one on the right wears jewelry. They are both barefoot. Most depictions of Roman women are more formal, with hair done and up, so the redhead on the left looks particularly unusual. It is thought that the mosaic was made by North African artists because some of the tesserae (pieces for the mosaic) are African.
The top is a band, probably a long piece of cotton or linen cloth wrapped around the breasts, rather than an actual bikini top. It is called a strophium. There are other possible names for it: fascia, fasciola, taenia, and mamillare. Its purpose was to hold the breasts and may also have been to compress them. The breast band was a normal, if optional, item in a woman's underwear. The same can not be said of the "bikini" bottom. The bottom loincloth-like piece is probably a subligar. A male athlete might wear a subligaculum. There may be no difference. The subligar/subligaculum was worn by athletes, slaves, and certain others who might otherwise wear too little clothing for Roman standards of modesty, but it was not a normal element of underwear, so far as we know. That is: normal Roman clothing included under-tunics and breast bands, but not underpants/drawers/briefs. So the silly Latin semper ubi sub ubi 'always where under where' may not apply to the Romans.
Whatever else, it is interesting to see the possibly idealized depiction of the shapes of the Roman women, with their firm, probably well-muscled legs and arms, broad hips, and small feet.
- "Roman Underwear Revisited," by Kelly Olson. The Classical World, Vol. 96, No. 2 (Winter, 2003), pp. 201-210.