The intrinsic spread of a configuration in 𝑅^{𝑑}

Conical equipartitions of mass distributions

“…Conditions1 and2 imply that thereis a naturalactionof thecircle groupSl on S suchthatF~S/ S 1 • Fromhereone deduces thatthespaceF of all pseudolines is homeomorphic to theprojectiveplanep2. This is known to hold for moregeneraltopologicalplanes,see [11] andthe references therein.Soit is reasonable to conjecturethatpart3 of Proposition4.3, which is thekey assumption for Theorem4.6,holdsfor general topologicalplaneswherethepseudolines arenot necessarilypiecewisesmooth. Definition 4.5.…”

“…Assumingthis definition of a pseudoline, a usualdefinition, see [11], of a topological plane Q is the following. 1. p2 is the usualprojectiveplaneand ,C = ,C(Q) is a topologicalspaceconsisting of simple,piecewisesmoothclosedcurves(pseudolines) in p2 playingtherole of lines in Q; 2. for anytwo distinctpointsp, q E p2,thereis a uniquepseudoline c(p, q) E ,C(Q) containingboth p andq; 3. every two distinct pseudolines/, l' E £(Q) intersectin exactlyonepoint where they intersecttransversally; 4. c(p, q) variescontinuouslyas a function of p andq; 5.…”

“…/ n /' variescontinuouslyas a function of / andI'. It was shownin [11] that every finite arrangement A of pseudolinesin p2 can be extendedto a topological plane. It is well known that there exist many examplesof nonstretchab/e arrangements of pseudolines, i.e., arrangements of pseudolinesnonisomorphic to any arrangement of straightlines.…”