We are given data α₁,..., αₘ and a set of points E = x₁,...,xₘ. We address the question of conditions ensuring the existence of a function f satisfying the interpolation conditions $f\left(x_i\right)={\alpha }_i$, i = 1,...,m, that is also n-convex on a set properly containing E. We consider both one-point extensions of E, and extensions to all of ℝ. We also determine bounds on the n-convex functions satisfying the above interpolation conditions.

We suggest a modification of the Pawłucki and Pleśniak method to construct a continuous linear extension operator by means of interpolation polynomials. As an illustration we present explicitly the extension operator for the space of Whitney functions given on the Cantor ternary set.

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.