Long ago Chazelle studied the polygonal version of your question, and obtained
a polynomial-time algorithm, polynomial in the number of vertices:

Bernard Chazelle, "The polygon containment problem,"
in *Advances in Computing Research*, Vol I: Computational
Geometry, (F.P. Preparata, ed.), JAI
Press, Greenwich, Connecticut (1983), 1–33.

If both polygons have $n$ vertices and neither can be assumed
convex, the polynomial is $O(n^7 \log n).$
If both are convex, then the complexity is reduced to $O(n^3)$.
Subsequently his results have been improved in various ways.
For example, the time complexity in the general case
was reduced by Avnaim & Boissonnat by a factor of $n$.^{1}
Google scholar lists
more than 100 papers that cite the original.

One application is
cartography: placing labels inside map regions:

^{
Fig.1(b) from Aonuma, H., Imai, H., Imai, K., & Tokuyama, T. "Maximin location of convex objects in a polygon and related dynamic Voronoi diagrams." In Proc. 6th Symposium Computational Geometry, 1990, pp. 225-234. ACM link.
}

^{1}Avnaim, Francis, and Jean Daniel Boissonnat. "Polygon placement under translation and rotation."

*Annual Symposium on Theoretical Aspects of Computer Science*. Springer Berlin Heidelberg, 1988.