Let \(B^ p_ n(r)=\{x^ n\in {\mathbb{R}}^ n:| x^ n|_ p\leq r\}\), where \(| x^ n|_ p=(\sum^{n}_{i=1}| x_ i|^ p)^{1/p}\), \(p>0\), \(S^ p_ n(r)=\{x^ n\in {\mathbb{R}}^ n:\;| x^ n_ p| =r\}.\) By \(U^ p_ n(r)\) we denote a uniform probability law on \(B^ p_ n(r)\), and by \(Q^ p_ n(r)\) a conditional probability law of \(x^ n\) with respect to \(U^ p_ n(r)\), under the condition \(| x^ n|_ p=r.\)
Let \(U^ p_{n,k}(r)\) and \(Q^ p_{n,k}(r)\) be probability laws of subvectors \(x^ k\) of \(x^ n\) with probability laws \(U^ p_ n(r)\) and \(Q^ p_ n(r)\), respectively, and let \(P^ p_ k(\sigma)\) be a probability law of a random vector \((\sigma Y_ 1,...,\sigma Y_ k)\), where the random variables \(Y_ i\) are independent with the density function \[ c_ p \exp \{-| x|^ p/p\},\quad c_ p^{- 1}=2\Gamma (1/p)p^{1/p-1}. \] The main result states that for \(n>k+p\) \[ \| Q^ p_{n,k}(r)-P^ p_ k(rn^{-1/p})\| \leq (k+2p)/(n-k- p),\quad p\geq 0, \] and \[ \| U^ p_{n,k}(r)-P^ p_ k(rn^{- 1/p})\| \leq (k+2p)/(n-k-p),\quad p\geq 1, \] where \(\| \cdot \|\) denotes the total variation.