Just to update this question with the revised AWS Lambda / API gateway concurrent limits:
AWS lambda is 1000 concurrent executions per region (thats across all functions) by standard, with a burst increase to 3000 (main regions) & 1000 (new / smaller regions). Details here: http://docs.aws.amazon.com/lambda/latest/dg/concurrent-executions.html
AWS API gateway is 1000 rps (requests per sec), I can’t see any burst provisioning for API gateway in the docs, ref: http://docs.aws.amazon.com/apigateway/latest/developerguide/limits.html#api-gateway-limits
So for 10K users you should be ok, as long as you don’t have them all hitting your app at the same time. Best bet is to monitor your requests per sec average and use that to guide your provisioning (monitoring is your friend here 
)