# Maximal mapping of a convex set to the unit disk

EDIT: To make my question more precise i think we can narrow it down to this. Say you have a simple polygon that includes the origin, that is completely contained in the unit disk, we can 'blow up' (description by Patrick) the polygon until it completely covers the disk. This 'blowing up' map can explicitly be given by the Schwarz–Christoffel mapping which is conformal. Now the similarity transform that scales (and maybe translates) the polygon so that it still is contained in the disk, but has maximal area, is trivially conformal. Can we somehow 'upgrade' the Schwarz–Christoffel map all the way to this similarity transform - maybe by intermediate maps that are all conformal and distort the lengths less but maybe fill less and less of the disk?

I am looking for an algorithm/theorem that helps me with the following: Given a convex set in the plane, I want to map it to the unit disk so that the image (of the convex set) has maximal area. With 'map to the unit disk' I mean that the image of the convex set is completely contained in the unit disk.

I know that for example for the Riemann mapping theorem you want a biholomorphic map - so I guess that also means you fill the whole disk. Whereas here the whole disk doesn't need to be filled. Does the Riemann-mapping theorem imply there will always be such a map so that the image has area 1? I would then be interested in restricting the map maybe all the way down to a homogeneous Euclidean transform. I am specifically interested in the case where the convex set is what you get from the intersection of half-planes (convex polygon). If there is a conformal such mapping does it mean that the image is a scaled/rotated version of the polygon maybe?

Anyway I am interested in the gritty details of accomplishing such a mapping (or of a similar sort).

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I notice that you have flagged a couple of your own questions as "not proper questions." Why? – mixedmath Dec 15 '12 at 23:55