English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this paper, we study continuous Kakeya line and needle configurations, ofboth the oriented and unoriented varieties, in connected Lie groups and someassociated homogenous spaces. These are the analogs of Kakeya line (needle)sets (subsets of $\mathbb{R}^n$ where it is possible to turn a line(respectively an interval of unit length) through all directions {\bfcontinuously, without repeating a "direction"}.) We show under some generalassumptions that any such continuous Kakeya line configuration set in aconnected Lie group must contain an open neighborhood of the identity, andhence must have positive Haar measure. In connected nilpotent Lie groups $G$,the only subspace of $G$ that contains such an unoriented line configuration isshown to be $G$ itself. Finally some similar questions in homogeneous spacesare addressed.  These questions were motivated by work of Z. Dvir in the finite fieldsetting.

Kakeya Configurations in Lie Groups and Homogeneous Spaces