 
Summary: ON INJECTIVITY OF COMBINATORIAL
RADON TRANSFORM OF ORDER FIVE
Tewodros Amdeberhan and Melkamu Zeleke
Abstract. In the present work, we give a proof of the injectivity of the combinatorial radon
transform of order ve.
The problem of determining members of a set by their sums of a xed order was posed by
Leo Moser and partially settled by Ewell, Fraenkel, Gordon, Selfridge, and Straus. Following
the notation of BL , the general problem can be stated in the following way.
For any given k;n 2 Z Z, with 2 k n, we choose arbitrarily an nset Xn =
fx1;x2;:::;xng then form the set Wk
nXn = f ig of all sums of k distinct elements of Xn and
ask:
Does there exist an nset X0n di erent from Xn giving rise to the same set of sums as does
Xn? More formally, we can describe the problem as follows:
De ne a mapping Wk
n from the set fXng of all nsets to the set of all
,n
k
sets by the rule:
