White shoes, especially sneakers, have a tendency to yellow over time. There are many myths as to why they yellow, but the basic theory is that direct sunlight breaks down the chemicals that are used to create a bright white sneaker. There are a few ways to reduce the yellowing, and luckily, it can be done cheaply and easily at home.
Remove the laces. You can set them aside and continue to use them eventually, but new laces are probably your best bet. Stuff the shoes with newspaper to hold their shape and make them easier to handle.
Start with one shoe at a time. Take the magic eraser and dip it in water. It should be damp, but not soaking wet. Rub the areas where yellowing has occurred with the magic eraser. It will lift some of the stains, but not all of them. You will be left with a slightly less noticeable yellow stain, but hopefully not too many.
Wipe the sneakers dry. Run your hand over the area to make sure it's absolutely dry.
Take the white shoe polish and cover the sneaker in smooth, even strokes. For large areas, use the applicator that comes with the shoe polish. Be careful around small edges. You may use a Q-tip or a makeup sponge to get into the small crevices.
When the shoes look good, leave them in a warm, dry place. Allow to completely dry and set. Do not place them in direct sunlight. Remove the newspaper when the sneakers are completely dry.