High-arched feet come with incorrect pronation and poor shock-absorption, which will set you up for trouble. If you aren't sure whether or not you have high arches, walk barefoot on wet grass, step on blacktop, and then examine your footprint. A normal arch has an imprint that connects the forefoot and heel with a wide band; a high arch has an imprint with no connecting band or a very narrow band between the forefoot and heel.
SOLUTION SHOE: The Adidas Adizero Adios 2 supports high arches by offering stability and allowing the front and back of the foot to transition independently when moving from heel to toe. For your heels, which absorb the most shock, the material underfoot iprovides superior cushioning. ($115, adidas.com)