A bounded subset of is a Kakeya set if it contains a unit line segment in every direction. Segment positions may vary with direction. Such a set need not have positive Lebesgue measure; the dimension problem concerns its Minkowski dimension or other notions of dimension.
The dimension assertion asks that every bounded Kakeya set in have full Minkowski dimension. The Kakeya maximal conjecture implies the stronger lower-dimension conclusion: its application to gives , hence for every .
A Kakeya tube is a neighborhood of transverse radius of a line segment of fixed unit length. Round cross-sections and comparable rectangular cross-sections are interchangeable up to fixed constants. The tube has Lebesgue measure comparable to in .
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A Kakeya set is a set of points in a Euclidean space (typically in two or higher dimensions) that has the property that a needle, or line segment, of unit length can be rotated freely within the set without leaving it. The classic example is the Kakeya set in the plane, which can be thought of as a bounded region that can contain a unit segment that can be rotated to cover all angles.