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Dense system of chords in the unit ball

Source: Miklos Schweitzer 2015, problem 1

March 25, 2016

real analysistopologycollege contests

Problem Statement

Let

K

be a closed subset of the closed unit ball in

\mathbb{R}^3

. Suppose there exists a family of chords

\Omega

of the unit sphere

S^2

, with the following property: for every

X,Y\in S^2

, there exist

X',Y'\in S^2

, as close to

X

and

K

correspondingly, as we want, such that

H\subset S^2

and

X'Y'

is disjoint from

K

. Verify that there exists a set

H\subset S^2

, such that

H

is dense in the unit sphere

S^2

, and the chords connecting any two points of

H

are disjoint from

K

.
EDIT: The statement fixed. See post #4

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