Palatine Chapel, Palermo, Sicily
Butler lists three saints of this name. The mosaic most likely represents the 5th-century martyr Eutropius of Constantinople. The cross in his hand is a common Orthodox symbol of martyrdom. He was a church lector (hence the tonsure), noted for his youth and beauty, which seem to be emphasized by the beardless face and rosy cheeks. The chapel's strong connections with the Eastern churches would make him more likely to be the subject of this mosaic than the other two saints of that name. His feast is on January 12. See Butler, I, 71-72, and Acta Sanctorum, January vol. I, 725-27.
Another martyred saint of this name is remembered as the first bishop of Saintes, a town in south-western France where he died in the third century. His feast is on April 30. See Butler, III, 199, and Acta Sanctorum, April vol. III, 733-44.
Finally, St. Eutropius of Orange was a bishop there in the 5th century. He was not a martyr. His feast day is May 27. See Butler, II, 405-406, and Acta Sanctorum, May vol. 6, 699-701.
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Photographed at the chapel by Richard Stracke, shared under Attribution-NonCommercial-ShareAlike license.