An infinitely variable diameter beverage sleeve includes a first surface configured to contact the container and extend around a circumference of the container, and a second surface configured to extend around the circumference of the container and face away from the container, and a securing device to hold the sleeve to the outside surface of the container. A first portion of the securing device is provided on one end of the sleeve on the first surface, and a second engaging portion of the securing means is provided as the second surface of the sleeve. The top edge of the sleeve is non-planar and the bottom edge of the sleeve is formed of a plurality of segments with at least two segments having a different radius of curvature.