A polished marble floor will scratch very easily compared to a harder stone like Granite. Tracking sand with shoes will scratch marble surfaces as with most natural stones. The reflectivity on a polished marble floor is caused by natural crystals in the stone. When sediment and grit are walked, dragged or scraped across the stone, the crystals become damaged and the surface is no longer even causing a loss in reflectivity and shine.
Therefore good barrier matting (rug or door mat) is essential as a preventive step. Keeping indoor slippers to change into is also recommended as outdoor grit can often get trapped in the treads of your shoes. Taking the above measures will help to prevent loss of shine in your New York marble, however, another important factor to consider is the possibility of staining.