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X \R(1 r\\ f /OO # / ( * ) --> e-*0 \B(e9 X)\ JB(t,x) for certain types of «-dimensional sets B(e9 x) shrinking to x as e -* 0. (\B(e9 x)\ is of course the Lebesgue measure of B(e9 x).) Some standard examples of sets B(e9 x) are balls with center x and radius e, and cubes with center x and diameter e. The problem of considering other sets besides balls and cubes for B(e9 x) received much attention in the 1930's. (See for example Buseman and Feller [1934].) This subject again seems to be attracting interest. One of the two main goals of this paper is an exposition of recent developments in which B(e9 x) is replaced by a lower dimensional set. 2

Problems in harmonic analysis related to curvature