Defining abstract duality as an order reversing involution, the only such transform on the class of convex functions is the Legendre transform. Restricting, however, the class to non-negative convex functions which are pinned down to equal zero at the origin, an essentially different duality transform appears. We will describe it, prove uniqueness results, and point out geometric interpretations and consequences, for example concerning the support and the Minkowski functional for log-concave functions. Based on joint work with V. Milman