Problem 988

If $P\subseteq S^2$ is a subset of the unit sphere then define the discrepancy

D(P) = \max_C \lvert \lvert C\cap P\rvert - \alpha_C \lvert P\rvert \rvert,

where the maximum is taken over all spherical caps $C$ , and $\alpha_C$ is the appropriately normalised measure of $C$ .

Is it true that

\min_{\lvert P\rvert=n}D(P)\to \infty

as $n\to \infty$ ?

Source: erdosproblems.com/988

Discrepancy

Wiki page

Status

SolvedProved

An accepted solution exists. The statement is true.

SOLVED, the site's label: Schmidt's 1969 theorem, refereed in Inventiones Mathematicae, answers the question yes in every dimension. The derived standing, solved and proved, is more specific than the site's label, which names no polarity: the accepted claim on the claim page (Schmidt, 1969) proves the statement.

Problem 988

Status

Proof claims

Discussion