10.4 Continuous Extensions, Part I

We give the convex and concave closures for a set function f. We show that they merit their name: the convex closure is indeed a convex continuous extension of f, and similarly, the concave closure is concave.

These two may be hard to evaluate.

This motivates the definition of the Lovasz extension, which is easy to evaluate.

We show that the Lovasz extension coincides with the convex closure if and only if f is submodular.

[ElIrracional]

In your example, the sum of lambdas is 1/2. Why can we claim that the sum of lambdas is always 1?