Minima and Saddles in the MLS Surface Definition Nina Amenta Summary: Minima and Saddles in the MLS Surface Definition Nina Amenta July 9, 2008 Shlomo Gortler and Zachary Abel (a Harvard undergrad) point out (personal communication) a problem in the proof of Claim 1 in our paper, "Defining Point-Set Surfaces" [1]. The Claim concerns the MLS surface [2], which is defined in terms of the MLS energy function eMLS(y, a), a function on the space IR3 ×P2. We first define the set Jx of point-direction pairs {(y, a) a = (y-x)/len(y-x)}, that is, a is the unit direction vector from x to y. Then Levin's definition can be stated: Definition: A point x belongs to the MLS surface if and only if x is a local minimum of eMLS(y, a) restricted to the set Jx. Our Claim was: Claim 1 The MLS surface consists of the points for which n(x) is well-defined, and for which x arglocalminyLx,n(x) eMLS(y, n(x)) (1) Here x is a point in IR3 , n(x) is the unit direction vector minimizing eMLS(x, a) over all a, and Lx,n(x) is the line through x with direction n(x). This Claim is an if-and-only-if statement; unfortunately only one direction is true. A point of the MLS surface does indeed satisfy Equation (1), but as Shlomo and Zachary point out there may