I haven't studied the notion of Jerusalem being the center of the world, but perhaps some of it might be pious belief.  Not sure what to do with that when the religious center of the world shifted to Rome.  Perhaps there's some notion related to the fact that Our Lord ascended into Heaven and that therefore the entry through the firmament to the throne of God would have to be above there somewhere.  There may of course be several entry points.  It's hard to say, really.  This is where real science and exploration (that isn't driven by an agenda) would come in.

This is some of the information I've collected regarding Jerusalem as the center of the world.  It appears collectively persuasive but would certainly turn the common view upside down.  I agree it would be nice to get some hard data.  The first account is from Andrew Dickson White's History of the Warfare of Science and Theology, and the next section is Anne Catherine Emmerich and the third is St. Hildegard maps.   
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There then stood in Germany, in those first years of the eight century, one of the greatest and noblest of men, --St. Boniface.  His learning was of the best then known. In labours he was a worthy successor of the apostles; is genius for Christian work made him unwillingly primate of Germany; his devotion to duty led him willingly to martyrdom.  There sat too, at that time, on the papal throne a great Christian statesman--Pope Zachary.  Boniface immediately declared against the revival of such a heresy as the doctrine of the antipodes; he stigmatized it as an assertion that there are men beyond the reach of the appointed means of salvation; he attacked Virgil, and called on Pope Zachary for aid.  Pg 105

The Pope, as the infallible teacher of Christendom, made a strong response. He cited passages from the book of Job and the widsom of Solomon against the doctrine of the antipodes; he declared it "perverse, iniquitous, and against Virgil's own soul," and indicated a purpose of driving him from his bishopric.  106

Warfare between science and theology White

The great authority of Augustine, and the cogency of his scriptural argument, held the Church firmly against the doctrine of the antipodes; all schools of interpretation were now agreed--the followers of the allegorical tendencies of Alexandria, the strictly literals exegetes of Syria, the more eclectic theologians of the West. For over a thousand years it was held in the Church, "always, everywhere, and by all," that there could not be human beings on the opposite sides of the earth, even if the earth had opposite sides; and, when attacked by gainsayers the great mass of true believers, from the fourth century to the fifteenth, simply used that opiate which had so soothing an effect on John Henry Newman in the nineteenth century--securus judicat orbis terrarum.  pg 104


The book of Ezekiel speaks of Jerusalem as in the middle of the earth, and all other parts of the world as set around the holy city.  Throughout the "ages of faith" this was very generally accepted as the direct revelation from the Almighty regarding the earth's form.  St. Jerome, the greatest authority of the early Church upon the Bible, declared, on the strength of this utterance of the prophet, that Jerusalem could be nowhere but at the earth's center; in the ninth century Archbishop Rabanus Maurus reiterated the same argument; in the eleventh century Hugh of St. Victor gave to the doctrine another scriptural demonstration; and Pope Urban, in his great sermon at Clermont urging the Franks to the crusade, declared, "Jerusalem is the middle point of the earth"; in the thirteenth century and ecclesiastical writer much in vogue, the monk Caesarious of Heisterbach declared, "As the heart in the midst of the body, so is Jerusalem situated in the midst of our in habited earth,--so it was that Christ was crucified at the center of the earth."  Dante accepted this view of Jerusalem as a certainty, wedding it to immortal verse: and in the pious book of ascribed to Sir John Mandeville, so widely read in the Middle Ages, it is declared that Jerusalem is at the center of the world, and that a spear standing erect at the Holy Sepulchre casts no shadow at the equinox.

     (30) For beliefs of various nations of antiquity that the earth's center
was in their most sacred place, see citations from Maspero, Charton,
Sayce, and others in Lethaby, Architecture, Mysticism, and Myth, chap.
iv. As to the Greeks, we have typical statements in the Eumenides of
Aeschylus, where the stone in the altar at Delphi is repeatedly called
"the earth's navel"—which is precisely the expression used regarding
Jerusalem in the Septuagint translation of Ezekiel (see below). The
proof texts on which the mediaeval geographers mainly relied as to the
form of the earth were Ezekiel v, 5, and xxxviii, 12. The progress
of geographical knowledge evidently caused them to be softened down
somewhat in our King James's version; but the first of them reads, in
the Vulgate, "Ista est Hierusalem, in medio gentium posui eam et in
circuitu ejus terrae"; and the second reads, in the Vulgate, "in medio
terrae," and in the Septuagint, [Greek]. That the literal centre of the
earth was understood, see proof in St. Jerome, Commentat. in Ezekiel,
lib. ii; and for general proof, see Leopardi, Saggio sopra gli errori
popolari degli antichi, pp. 207, 208. For Rabanus Maurus, see his De
Universo, lib. xii, cap. 4, in Migne, tome cxi, p. 339. For Hugh of
St. Victor, se his De Situ Terrarum, cap. ii. For Dante's belief, see
Inferno, canto xxxiv, 112-115:


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Anne Catherine Emmerich wrote:
... I learned also that the prophet having related what happened to him, the spot received the name of Calvary. Finally, I saw that the Cross of Jesus was placed vertically over the skull of Adam. 
I was informed that this spot was the exact center of the earth; and at the same time I was shown the numbers and measures proper to every country, but I have forgotten them, individually as well as in general. Yet I have seen this center from above, and as it were from a bird's-eye view. In that way a person sees far more clearly than on a map all the different countries, mountains, deserts, seas, rivers, towns, and even the smallest places, whether distant or near at hand.


 Hildegard's map with Jerusalem at the center. 

One interesting thought here is that it does seem likely that the continents were in fact once more connected than they are now.  You can see, for instance, most obviously, how South America fits with Africa like some puzzle.  We know that the lands did not separate "millions of years ago", as modern science claims, but I wonder if this may have occurred as a consequence of the Flood.  Or did God perhaps separate the lands at the time of the Tower of Babel, to scatter the different peoples?  Interesting line of inquiry.  This looks similar to that map in the posting from Tradman about St. Hildegard.

I stumbled upon this on GAB.

 
Yup.  Impossible to circuмnavigate "Antarctica" -- because it isn't a continent, but an ice wall surrounding the terra cognita on this flat world.

Well… I have made a conscious effort to eliminate "global" from my vocabulary.

Yup, me too.

More likely that things changed when the fountains of the deep erupted at the Flood.

Yes, makes sense.  When the fountains of the deep erupted, those could have pushed the continents apart.

Any mariners out there who could just sail east from NY to the next longitude line, do the equivalent in the southern hemisphere (Buenos Aires area), and track the time/distance?