Are there any sports sandals that provide enough support for running?

I run 10-15 miles a week, and I find good shoes an absolute necessity. I replace them several times a year. I’m no expert on the state the art – I buy the cheapest name-brand models I can find in the discount stores, which run me around $40 a pair.

I also have owned numerous sports sandals that were comfortable and good for long walks. But when I tried to run in them, I always got nasty shin and and ankle pain. (I usually run on either turf or a treadmill, and I weigh around 170 pounds. I do three miles in under 21 minutes).

I travel a lot for work, and go on a lot of long sightseeing walks on days off. Sports sandals are perfect for this (at least in warm weather), but a running-quality sports sandal would let me pack one less pair of shoes in my dinky suitcase.

Could someone advise if there is such a thing?