White color is associated with reflected (not absorbed) light. White paint usually includes a titanium oxide component, whose absorption is in UV.

The difference between a shiny surface and a Lambertian surface is its roughness, if light is reflected in a collective manner it look shiny. Any smooth surface is shiny given a grazing angle.

A mirror is usually coated with a metal surface, silver for example. Silver's metallic behavior makes most of the light reflected back (about %95 percent reflection, which is why it has a greyish tint), however it becomes transparent for UV light.

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