Using lowest latency AWS location automatically via Amazon Route 53:
Meanwhile, AWS is providing a useful map illustrating their worldwide Infrastructure for a fast assessment, alongside respective details like e.g. a number of availability zones and the API endpoint.
More significantly though, AWS has just declared the geographic DNS support, see the introductory post Multi-Region Latency Based Routing now Available for AWS, which is making available the same latency based routing technology that powers Amazon CloudFront to users of Amazon EC2, Elastic Load levelling, and more.
So just in case your environment is comprised of an Auto Scaling EC2 Instances architecture already, simply applying this latency based routing should solve your problem automatically.
While the use case clearly targets offerings spawning multiple AWS regions, the sophisticated options around Latency based Routing and Weighted Round Robin Record Sets may enable you to determine the desired information more easily yourself as well.