given a set  of  points in the plane and a constant ,-center problem is to find two closed disks which each covers the whole , the diameter of the bigger one is minimized, and the distance of the two centers is at least . constrained -center problem is the -center problem in which the centers are forced to lie on a given line . in this paper, we first introduce -center problem and its constrain...

In this paper, we generalize White's regularity and structure theory for mean-convex mean curvature flow [45, 46, 48] to the setting with free boundary. A major new challenge in boundary is derive an a priori bound ratio between norm of second fundamental form curvature. We establish such via maximum principle triple-approximation scheme, which combines ideas from Edelen [9], Haslhofer-Hershkov...

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