In 1944 the Duke and Duchess of Windsor stayed in the Waldorf Astoria
in New York. They presented a certain Roger Vergnes with a silver
cigarette case. Who is(was) this Vergner and what was his relation to
the ducal couple?

Hello.

Roger Vergnes was a New York City hairstylist.

Mr Vergnes' relationship to the Duchess of Windsor was professional. 
The Duchess of Windsor was one of Mr Vergnes' clients.


From The New York Times, January 27, 1948: 

'Mr Raymond... compared notes on the "new look" with Roger Vergnes,
whose shop at 36 East Fifty-seventh Street is patronized by the
Duchess of Windsor, Mrs Harrison Williamson and many stage stars."
source: "Hairdos Absorb British," The New York Times, January 27,
1948, page 29. Article available for purchase at the New York Times
Historical Archive:
http://pqasb.pqarchiver.com/nytimes/results.html?st=advanced&QryTxt=vergnes&By=&Title=&datetype=6&frommonth=01&fromday=27&fromyear=1948&tomonth=01&today=27&toyear=1948&restrict=articles&sortby=REVERSE_CHRON&x=37&y=12


Mr Vergnes' hair salon is also mentioned in this obituary from the
Providence Journal:
"She worked in New York City for Elizabeth Arden on Fifth Avenue, and
later managed the beauty salon of Roger Vergnes, then one of New
York's leading hairstylists."
http://www.projo.com/obituaries/content/projo_20020502_obit14.b412a.html

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search strategy:
google: "roger vergnes"
new york times historical archive: "roger vergnes"

Thanks.