1. Nira Dyn Tel-Aviv University joint work with Shay Kels

2. Points in a domain in orData: subsets of No embedding of sets into a linear space Applications in optimization, math. economics, computer vision and more An interesting mathematical question Assuming that the data are sampled from an SVF, we aim to define an approximant over the whole domain Many embedding of sets into a metric space

3. The “red” and the “green” sets are close in the symmetric difference metric, but are apart in the Hausdorff metric. The SVF is Hölder (Lipschitz) in and discontinuous in. Approximation results in apply to a wider class of functions. Our results are obtained in the metric.

4. The refinement rules are expressed by binary averages of numbers Averages of numbers are replaced by a new average of two sets – “the measure average” Spline subdivision schemes (e.g., the Chaikin scheme ) are adapted through the Lane-Riesenfeld algorithms The 4-point scheme is adapted by expressing the refinement rule as 3 binary averages, with positive and negative weights. 1 8 8 1 1 1

5. For metric property measure property

6. Set-valued spline schemes converge to Lipschitz continuous SVFs and approximate Hölder- SVFs with the order. The set-valued 4-point scheme converges to a continuous SVF for w<1/8, approximates Hölder- SVFs with the error of.

12. Given a collection of sets and non-negative weights, how to define a weighted average? We define a special partition of the union to disjoint sets: Each element of the partition consists of all points that belong to all sets for some but not to any other set.

13. To each element of the partition correspond the weights The partition average is obtained by taking a subset of each element of the partition with the measure as the sum of the corresponding weights

14. for any permutation of indices Extension of the metric property, important for approximation Commutativity The measure property

15. Families of positive operators for real-valued functions:,, Adaptation to SVFs with the partition average: Many well-known approximation operators can be adapted to SVFs, e.g., Bernstein, Schoenberg spline, piecewise linear. Since the partition average is commutative, the extension to operators for multivariate functions is straightforward. Average of numbers is replaced by the partition average of sets

16. If a family of positive operators for real-valued functions uniformly approximates continuous functions (as ), so does the corresponding family of operators for SVFs. If a family of positive operators for real-valued functions approximates Hölder- functions with the error, then the corresponding family of operators for SVFs approximates Hölder- SVFs with the error: with the norm of the partition