INSCRIBED POLYGONS AND FIXED-POINTS OF HOMEOMORPHISMS ON THE CIRCLE

We consider the problem of the inscription in a convex body of n-gons (n > 3) with edges in prescribed directions, generalizing a previous result by H. Cramer and A. Nemeth for n = 3. The method we give has suitable applications to problems of inscription of n-gons in general. We consider the types of convex bounded n-gons with edges having prescribed directions and give a formula for their number. For n = 4,5 necessary and sufficient conditions are provided for the existence of convex types of n-gons inscribable in the circle and for n = 5 we give a choice of directions such that no corresponding inscribed n-gon is convex.