Minimal Sublinear Representations of Convex Sets

Amitabh Basu, John's Hopkins University IAM Seminar
March 21, 2019 12:30 pm ESB 4133

It is well-known that a closed convex set C containing the origin in its interior can be represented as the 1-sublevel set of its gauge function. If the set C is compact, then the gauge is the *unique* sublinear function whose 1-sublevel coincides with C. However, if C is not compact, there can be multiple different sublinear functions whose 1-sublevels coincide with C. We call any such function a sublinear representation of C. It is not hard to see that the gauge of C is the largest sublinear representation of C, with respect to pointwise dominance. We show that there is a unique smallest sublinear representation f^* of C, i.e., f* <= f for any other sublinear representation f of C. The gauge, which is the largest sublinear representation of C, is well-known to be equal to the support function of the polar of C. We associate the notion of a “prepolar” with other sublinear representations and show that the geometric analog of the smallest sublinear representation is the concept of the smallest “prepolar”, with respect to set inclusion. This smallest “prepolar” has an explicit description, just like the classical polar.

Sushi served for lunch.