Ratios of affine functions

Aratio of affine functions is a function which can be expressed as the ratio of a vector valued affine function and a scalar affine functional. The purpose of this note is to examine properties of sets which are preserved under images and inverse images of such functions. Specifically, we show that images and inverse images of convex sets under such functions are convex sets. Also, images of bounded, convex polytopes under such functions are bounded, convex polytopes. In addition, we provide sufficient conditions under which the extreme points of images of convex sets are images of extreme points of the underlying domains. Of course, this result is useful when one wishes to maximize a convex function over a corresponding set. The above assertions are well known for affine functions. Applications of the results include a problem that concerns the control of stochastic eigenvectors of stochastic matrices.