Stuart Walker, 28, from Cumnock, was found dead with "horrible injuries" near Caponacre Industrial Estate, Cumnock, just before 05:00 on Saturday.
It is thought he had been tied to a lamppost, beaten and burned.
Following reports Mr Walker may have been targeted due to his sexuality, Strathclyde Police said they were looking into all aspects of his life.
Det Insp John Hogg, of Strathclyde Police, said Mr Walker was last seen alive about two hours before he was killed.
He said: "Stuart had been out with friends in the Cumnock area earlier during the night and was last seen alive by a family friend near to the fire station in Glaisnock Street around 2:30am on Saturday morning - nearly two-and-a-half hours before he was found.
"It is imperative that we find out where he was between 2.30am and 4.50am, who he was with and why this happened to him.
"From our inquiries so far, we understand that there may have been a number of house parties in the nearby Netherthird housing estate in the early hours of the morning.
"At this time we do not know if these parties are linked to our investigation or not, so, again, any information on that is important."
Adam Ingram, the SNP MSP for Carrick, Cumnock and Doon Valley, said Mr Walker's death had shaken the local community.
He told BBC Radio's Good Morning Scotland programme: "People are shocked and horrified.
"Stuart was a very nice young man, very popular and well-known within the Cumnock community, and their thoughts and hearts are going out to the family at the moment.
"It's shocking and horrifying, and it's come completely out of the blue. It's clearly shaken the local community."
Mr Ingram said he understood that vigils had been taking place in the area.
He added: "Stuart was very well-known and popular in the area, and I understand there have been many visits to the family home."
Family and friends described Mr walker as a "true friend and gentleman".
Hundreds of tributes have also been posted on Facebook.
A post from Karen Nicol on a page called RIP Stuart Walker, which almost 4,500 people have joined, read: "Life is so unfair. A genuine guy who always had a smile, a wave or a big cheery hello for everyone. Cumnock an emptier place without you for lots of people."
Anne Dunnachie wrote: "Every once in a while we are lucky enough to have known someone like you Stuart. A very special person...the world lost a wonderful person yesterday."