A kylix is a footed, two-handled, Greek drinking vessel. The red-figure technique started in the 6th century as a revolutionary pottery-painting technique. The name Penthesilea Painter was given to the anonymous artist who painted a particular bowl in which the inside depicts Achilles slaying Penthesilea.
Ganymede is known as the cup-bearer of the gods. Ganymede had been a mortal prince of Troy when his great beauty caught the eye of Zeus (Jupiter).
When Zeus kidnapped the most beautiful of mortals, the Trojan prince Ganymede, from Mt. Ida (where Paris of Troy was later a shepherd and where Zeus had been raised in safety from his father), Zeus paid off Ganymede's father with immortal horses. Ganymede's father was King Tros, the eponymous founder of Troy. Ganymede replaced Hebe as cup-bearer for the gods after Hercules married her.
Galileo discovered the bright moon of Jupiter which we know of as Ganymede. In Greek mythology, Ganymede was made immortal when Zeus took him to Mt. Olympus, so it's appropriate that his name should be given to a bright object that is forever in Jupiter's orbit.
On Ganymede, from Vergil's Aeneid Book V (Dryden translation):
There Ganymede is wrought with living art,
Chasing thro' Ida's groves the trembling hart:
Breathless he seems, yet eager to pursue;
When from aloft descends, in open view,
The bird of Jove, and, sousing on his prey,
With crooked talons bears the boy away.
In vain, with lifted hands and gazing eyes,
His guards behold him soaring thro' the skies,
And dogs pursue his flight with imitated cries.