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I know that magic squares exist: Summing over every row or column and diagonal one gets the same sum. My question is whether it is possible to generalize magic squares in such a way that the numbers are real and instead of summing over rows and diagonals one integrates over linear hypersurfaces and retrieves the same number. Has this been done already? How can I generate a family of distributions which satisfy this?

eeqesri
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    What do you mean by linear hypersurfaces? – anon Aug 10 '19 at 06:41
  • @runway44 Perhaps should have elaborated more on this: Say you have a 2D Box. By linear hypersurfaces I mean straight lines which intersect the box. – eeqesri Aug 10 '19 at 09:06
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    In the case of a discrete grid (so, usual magic squares) we don't consider sums over grid points on just any line, only certain ones, so arguably this generalization is a little different. If you consider only continuous functions, or $L^2$-integrable ones, or something like that, then presumably the only solutions are constant functions. A related topic in that case would be the Radon transform. – anon Aug 10 '19 at 09:33
  • @runway44 Yes the generalisation would be different as you said. But I don't understand how constant functions would be the only solution. Because different lines will intersect at different points and will also have different lengths. So a constant function won't satisfy the constraint. The Radon Transform looks interesting; I have to read up more on that. – eeqesri Aug 10 '19 at 09:54
  • Ah, I was thinking of normalized integrals over line segments. I guess if they're not normalized by line segment lengths then constant functions wouldn't be solutions. As your line segments get closer to the box's corner, they get smaller in length, approaching zero, so these functions would have to be unbounded near the corners... – anon Aug 10 '19 at 10:07
  • @runway44 initially I didn't understand, but yes you're right. The function would shoot up as the segment approaches the corner. – eeqesri Aug 10 '19 at 10:52

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