On the volume of non-central sections of a cube

Author name not available (Why is that?)

Abstract: Let

Q_{n}

be the cube of side length one centered at the origin in

m a t h b b R^{n}

, and let

F

be an affine

(n - d)

-dimensional subspace of

m a t h b b R^{n}

having distance to the origin less than or equal to

f r a c 12

, where

0 < d < n

. We show that the

(n - d)

-dimensional volume of the section

Q_{n} c a p F

is bounded below by a value

c (d)

depending only on the codimension

d

but not on the ambient dimension

n

or a particular subspace

F

. In the case of hyperplanes,

d = 1

, we show that

c (1) = f r a c 117

is a possible choice. We also consider a complex analogue of this problem for a hyperplane section of the polydisc.

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