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The Kakeya conjecture is generally formulated as one the followingstatements: every compact/Borel/arbitrary subset of ${\mathbb R}^n$ thatcontains a (unit) line segment in every direction has Hausdorff dimension $n$;or, sometimes, that every closed/Borel/arbitrary subset of ${\mathbb R}^n$ thatcontains a full line in every direction has Hausdorff dimension $n$. Thesestatements are generally expected to be equivalent. Moreover, the conditionthat the set contains a line (segment) in every direction is often relaxed byrequiring a line (segment) for a "large" set of directions only, where largecould mean a set of positive $(n-1)$-dimensional Lebesgue measure.  Here we prove that all the above forms of the Kakeya conjecture are indeedequivalent. In fact, we prove that there exist $d\le n$ and a compact subset$C$ of ${\mathbb R}^n$ of Hausdorff dimension $d$ that contains a unit linesegment in every direction (and also a closed set of dimension $d$ thatcontains a line in every direction) such that every subset $S$ of ${\mathbbR}^n$ that contains a line segment in every direction of a set of Hausdorffdimension $n-1$, must have dimension at least $d$.  We also obtain results on the duality of Hausdorff and packing dimensions viaadditive complements: For any non-empty Borel set $A$ of ${\mathbb R}^n$ weshow that  (1) the Hausdorff dimension of $A$ can be obtained as $n-p$, where $p$ is theinfimum of the packing dimension of those Borel subsets $B$ of ${\mathbb R}^n$for which $A+B={\mathbb R}^n$; and  (2) the packing dimension of $A$ can be obtained as $n-h$, where $h$ is theinfimum of the Hausdorff dimension of those Borel subsets $B$ of ${\mathbbR}^n$ for which $A+B={\mathbb R}^n$.

Equivalences between different forms of the Kakeya conjecture and duality of Hausdorff and packing dimensions for additive complements