I believe this is so, yes.
I think the advantage to minimizing distance-squared is that there exist semi-nice formulas to calculate the moment of inertia of a plane polygon in terms of its vertex positions, but I’m not sure that any such formulas exist for the mean absolute distance. (The formula in the link is derived from Green’s theorem somehow, but to do that you need to have a function with continuous partial derivatives in the region of integration, and the absolute distance function isn’t smooth at the origin.)
Anyhow, I managed to convince Mathematica to do this for me. (Code is below for anybody who’s interested. “fom” stands for “figure of merit.”) Outlying islands & exclaves were treated as being attached to the main part of the country by massless “struts”; as you can see, this penalized archipelago countries pretty heavily. Recall that the theoretical minimum figure of merit is 1/2π = 0.159155…; smaller numbers mean “rounder”.
Top twenty roundest countries (and figure of merit):[ol][li]Sierra Leone (0.163)[]Zimbabwe (0.168)[]Suriname (0.169)[]Poland (0.169)[]Ivory Coast (0.169)[]Swaziland (0.169)[]Nauru (0.171)[]Uruguay (0.171)[]Sudan (0.172; This one surprised me for a moment, but it’s actually pretty round now that South Sudan has separated)[]Egypt (0.172)[]Romania (0.173)[]Botswana (0.174)[]Macedonia (0.174)[]Vatican City (0.174)[]Belarus (0.175)[]Algeria (0.175)[]Gabon (0.176)[]Lesotho (0.176)[]Andorra (0.176)[]Ethiopia (0.177)[/ol][/li]
Top ten least round countries:[ol][li]Tuvalu (976.6)[]Maldives (924.8)[]Seychelles (872.3)[]Tonga (57.70)[]Bahamas (6.35)[]Cape Verde (4.18)[]São Tomé and Principe (3.97)[]Vanuatu (2.98)[]Comoros (2.28)[]Solomon Islands (2.26)[/ol][/li]
Top ten least round countries that are don’t consist of widely separated regions/islands (this is a judgement call on my part on what constitutes “widely”, of course):[ol][li]Chile (2.03; twelfth overall)[]Norway (0.899)[]Vietnam (0.894)[]Gambia (0.793)[]Japan (0.791)[]Cuba (0.783)[]Philippines (0.774)[]New Zealand (0.688)[]Israel (0.580)[*]Panama (0.498)[/ol][/li]
These results seem reasonable enough to me, so hopefully my brain will now stop bugging me to think about this problem.
fom[country_] :=
Module[{cname = country, pointlist, singlelist, basept, area,
centeredlist, centroid, moment},
pointlist = CountryData[cname, "Shape"][[1, 3, 1]];
basept = First[pointlist][[1]];
singlelist = pointlist[[1]];
For[i = 2, i <= Length[pointlist], i++,
singlelist =
Join[singlelist, {basept}, pointlist[li], {pointlist[[i, 1]]}]][/li] AppendTo[singlelist, basept];
area = 0;
For[i = 1, i < Length[singlelist], i++,
area = area +
1/2 (singlelist[[i, 1]] singlelist[[i + 1, 2]] -
singlelist[[i + 1, 1]] singlelist[[i, 2]])];
If[area < 0, singlelist = Reverse[singlelist]; area = Abs[area]];
centroid = {0, 0};
For[i = 1, i < Length[singlelist], i++,
centroid =
centroid +
1/6 {(singlelist[[i, 1]] +
singlelist[[i + 1,
1]]) (singlelist[[i, 1]] singlelist[[i + 1, 2]] -
singlelist[[i + 1, 1]] singlelist[[i, 2]]), (singlelist[[i,
2]] + singlelist[[i + 1,
2]]) (singlelist[[i, 1]] singlelist[[i + 1, 2]] -
singlelist[[i + 1, 1]] singlelist[[i, 2]])}];
centroid = centroid/area;
centeredlist = Map[(# - centroid) &, singlelist];
moment = 0;
For[i = 1, i < Length[centeredlist], i++,
moment =
moment +
1/12 (centeredlist[[i, 1]]^2 + centeredlist[[i, 2]]^2 +
centeredlist[[i, 1]] centeredlist[[i + 1, 1]] +
centeredlist[[i, 2]] centeredlist[[i + 1, 2]] +
centeredlist[[i + 1, 1]]^2 +
centeredlist[[i + 1,
2]]^2) (centeredlist[[i, 1]] centeredlist[[i + 1, 2]] -
centeredlist[[i + 1, 1]] centeredlist[[i, 2]])];
moment/area^2]
names = CountryData["Countries", "Name"];
roundnesses = Map[fom, names];
SortBy[Transpose[{names, roundnesses}], Last]