Area of A Box Containing Only One Point

[marcusl]

Imagine a rectangular grid of points with spacing (a,b) along the x and y directions, respectively, starting at the origin and filling all four quadrants. Center a rectangular box on the origin. The question is: what is the box of largest area that contains only the single point at the origin? The sides of the box can just touch other points, but no other point can be in its interior.

Trying some cases, I find:
a) trivially, the largest box whose sides are parallel to x and y has area A = 4ab.
b) if the box is tilted so its end touches a point on the line x = a (and, of course, the other end touches the mirror symmetric point at x = -a), the largest area is A = 4ab regardless of tilt.
c) if the box is long and skinny so it passes between two of the points located on the line x=a (and corresponding points on x=-a), A=4ab.

Well it looks like there's a pattern! Does anyone know of a general theorem?

This problem came up in the context of synthetic aperture radar imaging.

 

[Office_Shredder]

To start, I imagine you can assume the spacing is (1,1) and just scale the answer by a factor of ab at the end

 

[Xevarion]

Clearly there should be a lattice point on every side of the box, otherwise you'd be able to expand it more. Think about what happens when you rotate the box and you should be able to decide what angle it should be at too. That should be enough information I think...

 

[marcusl]

To start, I imagine you can assume the spacing is (1,1) and just scale the answer by a factor of ab at the end

Thanks. I had thought of doing this, but I didn't see any gains.

Clearly there should be a lattice point on every side of the box, otherwise you'd be able to expand it more. Think about what happens when you rotate the box and you should be able to decide what angle it should be at too. That should be enough information I think...

Yes, that works for each case (box extending to a point on x=+/-a, +/-2a, etc.). Was wondering if there is a general theorem at play.

 

[Xevarion]

http://en.wikipedia.org/wiki/Pick's_theorem

 

[marcusl]

EDIT: Revised message.

This is a beautiful result, but does not apply because the vertices do not fall on grid points for the case I'm considering (at least for any tilt angles except 0 and pi/2).

 

[andarrr]

This follows easily from Minkowski's theorem on convex bodies: Given a lattice L in R^n with a fundamental parallelotope of volume V, then any convex, symmetric region in R^n with volume >4V contains a nonzero point of the lattice.

 

[marcusl]

That's perfect! Thank you so much.

 

[symbolipoint]

Refering to the original simple question; you need to decide if you want to include the border or only the interior of the box.

No single box would contain only one theoretical geometry point, since a point has no size. Opposite sides of a box could never be one point distance apart. a side will always have infinitely points between itself and the reference point to be enclosed in the box.

... so, you have your choice of either 1 or 9. (there is a third choice, but not certain if you want that one).

 

[marcusl]

Hi s-p,

Sorry if my original post wasn't clear. The borders can touch other points but only one point (the origin) is allowed inside.