Convexity Conundrums

Various folks have requested copies of this article that originally appeared in the March 1997 issue of Risk Magazine. So, I have scanned in the article, at a very high resolution (300x300), in order to allow people to print it out, along with the mathematical proofs that went along with it. Because they are scanned at such high resolution, the images are very large (2528x3300), however the GIF compression makes the file sizes very reasonable. I recommend that the images be saved to your hard disk, and then printed on a laser printer, for maximum quality.

Please note that if you view some of the pages at a scaled (i.e. 50% zoom, etc.) size, the - sign sometimes disappears on your monitor, but not in the original GIF. To view the images properly on your monitor, I'd suggest using NO ZOOM, so that you see each pixel. A graphics viewer such as Irfanview (go to and search for "Irfanview") will produce excellent results.

Here is a description of all the pages: