Smooth a convex polygonal shape so that it becomes as large as possible while retaining diameter

Question

Given a convex polygon, I am trying to grow its shape (as in "maximal area") while preserving its diameter. The diameter is defined as the length of the longest segment that can be placed within the polygon. Since the polygon is convex, I assume that this diameter can always be found by scanning all vertex pairs.

For example, given an equilateral triangle as an input polygon, the diameter of the triangle is the length of any edge; smoothing this would result in 3 circle segments as shown in the image

For arbitrary convex polygons, a very inefficient algorithm is to compute the intersection of the maximum-diameter radius circles centered on each polygon vertex; this is what I am currently using (in Java). Is there anything better? Any pseudo-code or pointer to algorithm will be appreciated.

Answer 1

Computing the intersection's simpler than it looks; all you actually need to do is determine the point that's equidistant from two neighbouring vertices; you'll end up with two points, but whichever is closest to the centre of the shape will almost certainly be the right one; It might even be guaranteed for convex polygons, but mathematical proofs aren't my strong point.