# Shortest Distance between 2 convex sets

• TPks
In summary, The conversation discusses the problem of finding a bridge between two sets, with given functions and constants, of shortest possible length. The approach is to use the theorem of separating hyperplane and devise a search technique based on finding a line segment that is perpendicular to the tangent planes of the surfaces where it intersects. The suggested method is to start with an arbitrary line segment and adjust its ends to make it more perpendicular to the tangent planes.

#### TPks

Hi, I hope someone can help me out with this problem:

Let set S be defined by (x in En :f(x) <=c}
f: En -> E1 is convex and differentiable and gradient of f(xo) is not 0 when f(x) = c. Let xo be a point such that f(xo) = c and let d = gradient at xo. Let lamda be any positive number and define x1 = xo+ lamda(d)
we know:
||x1-xo|| <= ||x1-x||
suppse we have 2 islands:
f(x,y) <= c1 and g(x,y) <= c2
where f and g are contibuously differentiable and strictly convex functions whose gradient vectors are non zero when f(x,y) = c1 and g(x,y)=c2

We need to build a bridge between the two islands of shortest possible length. Using the result of the theorem of separating hyperplane, devise a search technique to find the optimal bridge location.