You have many options to go with. I'd suggest you search on Stackoverflow or gis.stackexchange forums.

Some approaches discussed on SO and GISSE below.

If you approach this using coordinates (centroids or places) try FNN. The package have fast implementations of kd-tree and cover-tree.

Just to give you a basic example:

library(FNN)

X cbind(X, as.data.frame(get.knn(X[,-1], k=1)))

> n x y nn.index nn.dist

> 1 a 1.0 1 2 1.000000

> 2 b 2.0 1 3 0.500000

> 3 c 2.5 1 2 0.500000

> 4 d 4.0 2 3 1.802776

http://stackoverflow.com/questions/16448402/distance-of-point-feature-to-nearest-polygon-in-r

http://stackoverflow.com/questions/21977720/r-finding-closest-neighboring-point-and-number-of-neighbors-within-a-given-rad

http://stackoverflow.com/search?q=%5Br%5D+nearest+distance

http://gis.stackexchange.com/questions/83722/calculate-minimal-distance-between-polygons-in-r

This is an interesting question.  I'm assuming you want distances corrected for spherical geometry?  In which case you have to use 'great circle' distances.  I like the explanation found here:  http://www.movable-type.co.uk/scripts/latlong.html

Thank you gentlemen! 

Richard John Smith Try the package 'sf' https://github.com/r-spatial/sf It is ultrafast. There the function st_distance https://r-spatial.github.io/sf/reference/geos_measures.html

Best, Sergio