Minimum distance between two SuperEllipsoids.

Optimization method to determine the minimum distance (or max overlap) between two SuperEllipsoids?

What´s the minimum distance between two SuperEllipsoids? What's the maximum overlap between two SuperEllipsoids?
A proximity query contact detection between convex superellipsoids surfaces using its implicit equations.
The contact detection of two superellipsoids is formulated as a convex nonlinear constrained optimization problem that is solved using fmincon function, with an Interior Point method. The objective function to be minimized is the distance between both surfaces. The design constraints are the implicit superquadrics surfaces equations and operations between its normal vectors and the distance itself (several constraint sets can be selected). The contact points or the points that minimize the distance between the surfaces are the design variables.
The visualization of the surfaces, the initial estimate and the optimization solution for the minimum distance is provided.

References:
- Portal, Ricardo. Sousa, Luís. Dias, João. "Contact Detection between Convex Superquadric Surfaces". The Archive of Mechanical Engineering. Versita, Warsaw. LVII(2), pp. 165-186, DOI 10.2478/v10180-010-0009-8. 2010.

- Chakraborty, N., J. Peng, et al. (2008). "Proximity Queries Between Convex Objects: An Interior Point Approach for Implicit Surfaces." IEEE Transactions on Robotics: 211-220.