The Polar and Bipolar of a Convex Polytope
This Demonstration illustrates the concepts of the polar and the bipolar of a convex polytope in . A polytope is the convex hull of a finite set of points. (In two dimensions, polytopes are convex polygons.) Initially there are three points, which are the vertices of a polygon displayed in red, but you can increase their number with Ctrl+click (command+click on Mac OS X) anywhere within the displayed region. The other convex regions visible in the graphic are the polar of the polytope (blue) and the polar of the polar—the bipolar of the polytope (green). The polar of a subset of is the set . The polar of a convex polytope is also convex, but, in general, need not be bounded. The polar of a polytope is bounded (i.e. is a polytope) if and only if the polytope contains the origin in its interior.
This Demonstration can be used to illustrate the well-known properties of polytopes and their polars and bipolars (see ). In particular, both the polar and the bipolar of a polytope contain the origin; a polytope is always contained in its bipolar and coincides with it if it contains the origin. In general, the bipolar coincides with the convex hull of the polytope together with origin; see theorem 6.2 of .
 A. Brøndsted, An Introduction to Convex Polytopes, New York: Springer–Verlag, 1983.