Séance Séminaire

Séminaire ACSIOM

Borsuk's $k$-regular embeddings, Legendre interpolation and Chebyshev approximation.

The mapping $F:X\to R^m$ of compact space $X$ is called $k$-regular if preimage of any $(k-1)$-dimensional plane in $R^m$ has no more than $k$ points. 1-regular maps are exactly embeddings. K.Borsuk proposed the problem: For given natural numbers $n$ and $k$ estimate the minimal number $m(n,k)$ such that any $n$-dimensional compact space $X$ can be $k$-regularly embedded into $R^m$. V.Boltjanski proved that $m\leq nk+n+k$. We show that $m\geq nk+n+k$ for any odd $k$. $k$-regular maps are important due to the Rubinshtein theorem: For linearly independent functions $1,f_1,...,f_m:X\to R$ the following conditions are equivalent: 1) The mapping $F=(f_1,...,f_m):X\to R^m$ is $k$-regular. 2) For any points $x_0,...,x_k\in X$ and any numbers $t_0,...,t_k\in R$ there exist numbers $\alpha_0,...,\alpha_m$ such that $f(x_i)=t_i$, $i=0,...,k$ for function $f=\alpha_0+\alpha_1f_1+...\alpha_mf_m$. 3) For any continuous function $f:X\to R$ the set $A$ of best Chebyshev approximations in $L(1,f_1,...,f_m)$ has dimension $\leq m-k$.