To preface, this question might come off as a bit silly, but it would serve me well to understand the concepts, and to actually get a good answer for this.
How can I design an ideal metric for walking places. I know about taxi cab geometry, so I know how to do things for city streets, but what I am more concerned about is sunlight. I don't like the sun very much and burn pretty easily. Therefore, when I use heuristics when I walk place,s by default, I take routes that involve more shade. I was talking with a friend about how to make this into a metric. She has taken classes involving metrics and I have not, but I have a basic understanding of them. Basically I proposed a metric that said that distances in the shade are shortened by some constant (real) factor. The eventual conclusion was that if you define the distance to be the minimum distance over all possible paths, where distance in the sun is measured in the Euclidean metric and distance in the shade is measured by k*Euclidean metric where $0<k<1$. Why can't I make k infintesimal? I know this breaks the triangle inequality, but why? My intuituion says that making k infintesimal is equivalent to scaling the distance in the sunlight by a factor of infinity. Is there some weird non-standard analysis trick I can use to make this work? Additionally, I was also wondering if I could make something that makes going up stairs worse than going down. I was told this violates reflexivity, but again is there a trick around this? . Thanks so much.