Cut the clementines into quarters and squeeze their juice into a saucepan. Add the sugar, water and clementine flesh and bring to a boil over medium heat. Reduce the heat and simmer the mixture for 5 minutes. Pour into a strainer set over a measuring cup, pressing on the solids to remove as much liquid as possible. Discard the solids, cool the liquid to room temperature and pour into a pitcher. Add the cranberry juice, lime juice and vodka and stir well. Refrigerate until ready to serve. Put 6 or 8 martini glasses in the freezer.
To serve, pour the mixture into an ice-filled cocktail shaker and shake until very cold. Pour into the chilled martini glasses and garnish each with a clementine slice. Drink up and enjoy!