How many Capitals are closer to me than my own?

Question

A recent question in chat about "When you are in Aachen, Germany, five other countries capitals are closer than Berlin." triggered me to try to write something to find the cities automatically for whatever city I like.

At first, I get all UN countries and their respective capitals.

countries = CountryData["UN"];
capitals = Map[CountryData[#, "CapitalCity"] &, countries];
capitalCoord = 
  Select[Map[{#, CityData[#, "Coordinates"]} &, capitals], 
   NumericQ[#[[2, 1]]] &];

The "Select..."-part in capitalCoord is due to the fact, that some countries (before I only selected UN countries) did not have a listed capital. I just left it in, because this is only executed once.

Second is to define a reference city (here Aachen) and based on the found capitals above to get all distances to all capitals, sorting them, and show all closer and the own capital.

refCity = CityData[{"Aachen", "NorthRhineWestphalia", "Germany"}];
ref = GeoPosition[refCity];
erg = Map[{#[[1]], GeoDistance[ref, #[[2]]]} &, capitalCoord];
With[{l = Sort[erg, #1[[2]] < #2[[2]] &], 
   city = CountryData[CityData[refCity, "Country"], "CapitalCity"]}, 
  For[i = 1, i <= Length[countries], i++, 
   If[l[[i, 1]] == city, Break[]]]];
GeoGraphics[Insert[With[{l = Sort[erg, #1[[2]] < #2[[2]] &][[;; i]]},
   Table[GeoPath[{l[[j, 1]], refCity}], {j, i}]], Red, i], 
 GeoProjection -> "Robinson", 
 PlotLabel -> ToString[i - 1] <> " UN nation capitals are closer"]

Some examples (Red is always the own capital):

Aachen, Germany

There are actually seven capitals closer than Berlin. Then we were trying more fancy cities …

Vladivostok, Russia

Kashi, China

Unalaska, USA

Honolulu, USA

My question is now … again … what would be a more elegant way to do it?

If you can do it more fancy, please do. :)

Answer 1

Using GeoNearest:

capitals = Map[CountryData[#, "CapitalCity"] &, CountryData["UN"]];
nearest = GeoNearest[capitals, Here, 10]

And then use LengthWhile like this to answer the question in the title (the capital in my country is Stockholm):

I only printed the ten closest capitals for display purposes, you can use this:

nearest = GeoNearest[capitals, Here, All]

You can also use free-form input in place of CountryData:

There is also TakeWhile if you want to make a visualization. Here is a suggestion. I apologize for using images for code, but I like how entities look in notebooks:

Note that my answer to the question "how many capitals are closer to me than my own?" is short enough to be a one-liner, if using free-form input:

Answer 2

For efficiently retrieving nearby cities, use Nearest[] with a custom DistanceFunction:

capitals = Map[CountryData[#, "CapitalCity"] &, CountryData["UN"]];
locs = CityData[#, "Coordinates"] & /@ capitals;

nf = Nearest[locs -> capitals, DistanceFunction -> (QuantityMagnitude @* GeoDistance)];

Then:

CommonName /@ nf[CityData[{"Aachen", "NorthRhineWestphalia", "Germany"}, "Coordinates"], 7]
   {"Brussels", "Luxemburg", "Amsterdam", "Paris", "Bern", "London", "Vaduz"}

From that, just increase the value of the second argument until the last city returned is in the same country as the original city:

nearestCapitals[city_, opts___] := Module[{k = 0, res},
       While[res = nf[CityData[city, "Coordinates"], ++k]; 
             CityData[Last[res], "Country"] =!= CityData[city, "Country"]];
       GeoGraphics[{Tooltip[GeoMarker[city, "Color" -> Blue], CommonName[city]], 
                    MapAt[{Directive[Red, Thick], #} &, 
                          GeoPath[{city, #}, "Geodesic"] & /@ res, {-1}], 
                    Tooltip[GeoMarker[#], CommonName[#]] & /@ res}, opts, 
                   PlotLabel -> StringForm["`` UN capital cities are closer",
                                           Length[res] - 1]]]

nearestCapitals[Entity["City", {"Unalaska", "Alaska", "UnitedStates"}], 
                GeoBackground -> GeoStyling["CountryBorders"]]