Min-max and max-min identities are found for inner products on the boundaries of compact, convex sets whose interiors contain the origin. The identities resemble the minimax theorem but they are different from it. Specifically, the value of each min-max (or max-min) equals the value of a dual problem of the same type. Their solution sets can be characterized geometrically in terms of the enclosed convex sets and their polar sets. However, the solution sets need not be convex nor even connected.
