Hausdorff Measure - Relation With Hausdorff Dimension

Relation With Hausdorff Dimension

One of several possible equivalent definitions of the Hausdorff dimension is

\operatorname{dim}_{\mathrm{Haus}}(S)=\inf\{d\ge 0:H^d(S)=0\}=\sup\bigl(\{d\ge 0:H^d(S)=\infty\}\cup\{0\}\bigr),

where we take

Read more about this topic:  Hausdorff Measure

Famous quotes containing the words relation with, relation and/or dimension:

    “To criticize is to appreciate, to appropriate, to take intellectual possession, to establish in fine a relation with the criticized thing and to make it one’s own.”
    —Henry James (1843–1916)

    “The instincts of the ant are very unimportant, considered as the ant’s; but the moment a ray of relation is seen to extend from it to man, and the little drudge is seen to be a monitor, a little body with a mighty heart, then all its habits, even that said to be recently observed, that it never sleeps, become sublime.”
    —Ralph Waldo Emerson (1803–1882)

    “Le Corbusier was the sort of relentlessly rational intellectual that only France loves wholeheartedly, the logician who flies higher and higher in ever-decreasing circles until, with one last, utterly inevitable induction, he disappears up his own fundamental aperture and emerges in the fourth dimension as a needle-thin umber bird.”
    —Tom Wolfe (b. 1931)