In this paper, results motivated by problems coming from convex-geometry and from non-parametric statistics (learning theory) are obtained.
Let \(e_1,\dots,e_n\) be the standard basis in \(\mathbb{R}^n\) endowed with the canonical Euclidean structure, and let \(\{g_i\}_1^n\) be independent \(\mathcal{N}(0,1)\) Gaussian random variables. Let \(T\subset\mathbb{R}^n\), and consider the sets \(T_{\rho}=T\cap\rho B_2^n\), where \(B_2^n\) is the unit Euclidean ball.
The purpose of the paper is to look for bounds of \(l_{\ast}(T_{\rho}):=E\sup_{l\in T_{\rho}}\langle\sum_{i=1}^ng_ie_i,t\rangle\) as a function of \(\rho\).
Sharp bounds are established when \(T=B_p^n\) for \(1\leq p\leq\infty\) (namely, the unit ball of the \(l_p^n\)-norm, defined as \(\| x\| _p=\left(\sum_i| x_i| ^p\right)^{\frac{1}{p}}\) for \(p<\infty\), and \(\| x\| _{\infty}=\sup_i| x_i| \)), or \(T=B_{p\infty}^n\) for \(0<p\leq 1\) (namely the unit bal of the so-called weak \(l_p^n\)-norm). In fact, sharp bounds are also obtained when \(B_2^n\) is replaced by \(B_q^n\).
An application to statistics is presented in connection with the so-called approximate reconstruction problem. Also, applications to convex geometry are discussed by showing that the control of \(l_{\ast}(T_{\rho})\) allows the control of the diameter of \(k\)-codimensional sections of \(T\) for an appropriate \(k\), so that the main results of the paper have immediate consequences for the diameter of sections. Precise estimates are provided in some cases.