Let $H\subset \mathbb{R}^n$ be a $k$-dimensional affine space and suppose $vol_k(H\cap [-\frac{1}{2},+\frac{1}{2}]^n)>0$ then can one upper bound the ratio $$\frac{vol_k(H\cap [-\frac{1}{2},+\frac{1}{2}]^n\cap B(0,\sqrt{\alpha k}))}{vol_k(H\cap [-\frac{1}{2},+\frac{1}{2}]^n)}.$$ I conjecture that this is at most $(C\alpha)^{k/2}$ for some absolute constant $C$, because that's what come out of taking the axis parallel subspace $H=\{x\in \mathbb{R}^n: x_i=0, ~i>k\}$. I can show this is true for subspaces, the problem is proving it for affine spaces, because in that case the numerator is always the same for all $H$ and the denominator is at least 1.
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$\begingroup$ The upper bound is 1. Did you mean the lower bond? For fixed $k$, it goes to $0$ as $n\to\infty$. $\endgroup$– Anton PetruninApr 14, 2021 at 4:34
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1$\begingroup$ I want an upper bound better than 1, isn't it possible to get one? I conjecture one should be able to get an upper bound of $(C\alpha)^{k/2}$. $\endgroup$– user86558Apr 14, 2021 at 14:07
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