Starting Web Api 2 with OWIN

help you fix your problem An answer from my codeplex issue

Does that help I found something similar. My Configuration method was running but the Authentication middleware (OpenIdConnect in this instance) did not seem to be executing in the Owin pipeline when hosted as a child Web Application. It worked fine when hosted as a root website.
Before delving any deeper into debugging the pipeline itself I tried a couple of things that I have seen cause issues when using other token services (ADFS for one). And sure enough it turned out to be one of them.

app.UseOpenIdConnectAuthentication(new OpenIdConnectAuthenticationOptions
{
    Authority = "https://idserver/identity",
    ClientId = "Client",
    RedirectUri = "https://clients/mvcclientapp"
    ...
});

app.UseOpenIdConnectAuthentication(new OpenIdConnectAuthenticationOptions
{
    Authority = "https://idserver/identity",
    ClientId = "Client",
    RedirectUri = "https://clients/mvcclientapp/"
    ...
});

To fix the issue you can do Do not use the * in your proposed route template. It will cause conflicts. {prefix}/api/{controller}/{action} should work for your 1st and 3rd option. Optional placeholders are not allowed in the middle of route templates so you would have to register the api/{controller}/{action} separately.

config.Routes.MapHttpRoute(
    name: "PrefixedControllerAndAction", 
    routeTemplate: "{prefix}/api/{controller}/{action}"
); 

config.Routes.MapHttpRoute(
    name: "ControllerAndAction", 
    routeTemplate: "api/{controller}/{action}"
);