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For an $r$-tuple $(gamma_1,ldots,gamma_r)$ of special orthogonal $dtimes d$ matrices, we say the Euclidean $(d-1)$-dimensional sphere $S^{d-1}$ is $(gamma_1,ldots,gamma_r)$-divisible if there is a subset $Asubseteq S^{d-1}$ such that its translations by the rotations $gamma_1,ldots,gamma_r$ partition the sphere. Motivated by some old open questions of Mycielski and Wagon, we investigate the version of this notion where the set $A$ has to be measurable with respect to the spherical measure. Our main result shows that measurable divisibility is impossible for a generic (in various meanings) $r$-tuple of rotations. This is in stark contrast to the recent result of Conley, Marks and Unger which implies that, for every generic $r$-tuple, divisibility is possible with parts that have the property of Baire.
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Laczkovich proved that if bounded subsets $A$ and $B$ of $R^k$ have the same non-zero Lebesgue measure and the box dimension of the boundary of each set is less than $k$, then there is a partition of $A$ into finitely many parts that can be translated to form a partition of $B$. Here we show that it can be additionally required that each part is both Baire and Lebesgue measurable. As special cases, this gives measurable and translation-on
We prove that any convex body in the plane can be partitioned into $m$ convex parts of equal areas and perimeters for any integer $mge 2$; this result was previously known for prime powers $m=p^k$. We also discuss possible higher-dimensional generalizations and difficulties of extending our technique to equalizing more than one non-additive function.
Given an action of a group $Gamma$ on a measure space $Omega$, we provide a sufficient criterion under which two sets $A, Bsubseteq Omega$ are measurably equidecomposable, i.e., $A$ can be partitioned into finitely many measurable pieces which can be rearranged using the elements of $Gamma$ to form a partition of $B$. In particular, we prove that every bounded measurable subset of $R^n$, $nge 3$, with non-empty interior is measurably equidecomposable to a ball via isometries. The analogous result also holds for some other spaces, such as the sphere or the hyperbolic space of dimension $nge 2$.
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Consider a polygon P and all neighboring circles (circles going through three consecutive vertices of P). We say that a neighboring circle is extremal if it is empty (no vertices of P inside) or full (no vertices of P outside). It is well known that for any convex polygon there exist at least two empty and at least two full circles, i.e. at least four extremal circles. In 1990 Schatteman considered a generalization of this theorem for convex polytopes in d-dimensional Euclidean space. Namely, he claimed that there exist at least 2d extremal neighboring spheres. In this paper, we show that there are certain gaps in Schattemans proof, which is based on the Bruggesser-Mani shelling method. We show that using this method it is possible to prove that there are at least d+1 extremal neighboring spheres. However, the existence problem of 2d extremal neighboring spheres is still open.
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