"A case has been opened to investigate the ambassador's disappearance," Rio state police said in a statement.
The investigation is being led by the missing persons section of the homicide division in the Rio neighborhood of Baixada Fluminense, on the city's west side, it said.
According to "preliminary information," Amiridis was last seen on the night of December 26, it said.
Amiridis, 59, had been on vacation in the picturesque but crime-plagued city since December 21. He was due to fly back to Brasilia on January 9, a Greek embassy official told AFP.
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Brazilian media reports said the ambassador's wife had reported him missing after he left the apartment they were renting in the city of Nova Iguacu, outside Rio, and never came back.
She was reportedly unable to get in touch with him.
Amiridis was named ambassador this year. He had previously served as Greece's consul general in Rio from 2001 to 2004.
He served as Greece's ambassador to Libya from 2012 to 2016.
He is married and has a daughter, according to the embassy in Brasilia.
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