Could it be the cap off a (wooden?) rod which has been ground down?
Are the inner edges of the curve flat, as on the same plane?
From memory, there was a similar looking item on TNet some time ago, but I think it was bigger and I think it was identified as the cap for a horse buggy arm. Worn away where it rested and dragged on the ground - when the horse wasn't attached
Otherwise, first thought was pencil sharpener
Mike