I stumbled across the following assertion in an example in a paper I'm reading on chance constrained optimization. Although it seems intuitively true, I'm not quite sure how to approach this and would appreciate any hints/thoughts.
Let $\{\xi_i\}^{N}_{i=1}$ be $N$ independent random samples of a normally distributed random variable $\xi \sim N(0,I_d)$. Show that for each $p \in (0,1)$, there exists a large-enough dimension $d \in \mathbb{N}$ and a vector $x \in \mathbb{R}^d$ such that the following conditions are jointly satisfied with probability $\geq p$:
- $\lVert x \rVert_2 = 1$,
- $\xi^{\text{T}}_i x = 0$, $\:\forall i \in \{1,\cdots,N\}$,
- $e^{\text{T}}_d x \leq -\frac{0.1}{\sqrt{d}}$, where $e_d \in \mathbb{R}^d$ is a vector of ones.
I imagine that the above result would still hold if we were to consider $\xi \sim N(0,\Sigma)$ for a general covariance matrix $\Sigma$ and also require $e^{\text{T}}_d x \leq -\frac{C}{\sqrt{d}}$ for any given constant $C > 0$.
My thoughts: Clearly for $d > N$, we can satisfy the first two conditions by looking at a $(d-N)$-dimensional subspace of $\mathbb{R}^d$ and considering the vectors of unit norm in this subspace. All that remains to show is that for $d$ large, we can arrange $e_d$ to make an angle suitably bounded away from $90^{\circ}$ with some of the points on this subspace that are of unit norm. I can see that it is quite unlikely that any of the $\xi_i$s will be a scalar multiple of $e_d$ (therefore, we should be able to find an $x$ such that $e^{\text{T}}_d x < 0$), but I'm not sure how to approach the desired inequality.