Let $Y\subset l_{\infty }^{\left(n\right)}$ be a hyperplane and let $A\in ℒ\left(Y\right)$ be given. Denote $$\begin{array}{ccc}\hfill 𝒜=& \{L\in ℒ(l_{\infty }^{\left(n\right)},Y):L\mid Y=A\}\text{and}\hfill & \hfill {\lambda }_A=inf\{\parallel L\parallel :L\in 𝒜\}.\end{array}$$ In this paper the problem of calculating of the constant ${\lambda }_A$ is studied. We present a complete characterization of those $A\in ℒ\left(Y\right)$ for which ${\lambda }_A=\parallel A\parallel$. Next we consider the case ${\lambda }_A>\parallel A\parallel$. Finally some computer examples will be presented.

Let $S=\{x_1,\cdots ,x_n\}$ be a finite subset of a partially ordered set $P$. Let $f$ be an incidence function of $P$. Let $\left[f(x_i\wedge x_j)\right]$ denote the $n\times n$ matrix having $f$ evaluated at the meet $x_i\wedge x_j$ of $x_i$ and $x_j$ as its $i,j$-entry and $\left[f(x_i\vee x_j)\right]$ denote the $n\times n$ matrix having $f$ evaluated at the join $x_i\vee x_j$ of $x_i$ and $x_j$ as its $i,j$-entry. The set $S$ is said to be meet-closed if $x_i\wedge x_j\in S$ for all $1\le i,j\le n$. In this paper we get explicit combinatorial formulas for the determinants of matrices $\left[f(x_i\wedge x_j)\right]$ and $\left[f(x_i\vee x_j)\right]$ on any meet-closed set $S$. We also obtain necessary and sufficient conditions for...

Let T be the standard Cantor-Lebesgue function that maps the Cantor space $2^{\omega }$ onto the unit interval ⟨0,1⟩. We prove within ZFC that for every $X\subseteq 2^{\omega }$, X is meager additive in $2^{\omega }$ iff T(X) is meager additive in ⟨0,1⟩. As a consequence, we deduce that the cartesian product of meager additive sets in ℝ remains meager additive in ℝ × ℝ. In this note, we also study the relationship between null additive sets in $2^{\omega }$ and ℝ.