A set with finite curvature and projections of zero length
A compact set \(E\) in the plane has finite Melnikov (Menger) curvature if it supports a measure \(\mu\) such that \(\mu(\Delta)\leq |\Delta|\) for a very closed disk \(\Delta\) of diameter \(|\Delta|\) and such that \[ \int_E\int_E\int_E R(x,y,z)^{-2} d\mu(x) d\mu(y) d\mu(z)<\infty, \] where \(R(x,y,z)\) is the radius of the circle that passes through the points \(x,y,z\). Recently \textit{G. David} and \textit{J. C. Léger} [Ann. Math. 149, 831-869 (1999)] have proved that a compact planar set of finite length has finite Melnikov curvature if and only if it projects onto a set of positive length in almost all directions. The authors of the paper under review present an example showing that this theorem fails without the finite length assumption. The construction is similar to one by \textit{M. A. Martin} and \textit{P. Mattila} [Trans. Am. Math. Soc. 305, 293-315 (1988; Zbl 0643.28009)]. The Melnikov curvature has played an important role in the old open problem of characterizing the removable sets for bounded analytic functions. The authors remark that their example shows that the finite length assumption is necessary in Vitushkin's conjecture [recently proved by \textit{G. David}, Rev. Mat. Iberoam. 14, No. 2, 369-479 (1998; Zbl 0913.30012)]. This fact was known because of a more complicated example by \textit{P. W. Jones} and \textit{T. Murai} [Pac. J. Math. 133, No. 1, 99-114 (1988; Zbl 0653.30016)].
scientific article; zbMATH DE number 1497292
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