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nLab > nLab General Discussions: star-shaped neighborhood

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    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 19th 2015
    • (edited Sep 19th 2015)

    In star shaped region (titled “star-shaped neighborhood), there is insistence in the definition on the region being open in the ambient affine/vector space. That seems like a bad decision to me. For example, the statement in convex set “every convex set is star-shaped” is incorrect under that definition.

    It seems to me the page should be renamed “star domain” (after Wikipedia), and have the simpler definition that there exists a point x 0∈Dx_0 \in D such that if xx belongs to DD, then so does tx 0+(1−t)xt x_0 + (1-t)x for all t∈[0,1]t \in [0, 1]. Agree?

    • CommentRowNumber2.
    • CommentAuthorspitters
    • CommentTimeSep 19th 2015

    I think you are right. I tried to find it in pi base, but alas.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeSep 21st 2015

    I don’t disagree! :-)

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 21st 2015

    I changed the name to star domain and added a bit more e.g., the observation that open stars provide a good open cover of a simplicial complex.

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