Interpolation of toric varieties

Let $X\subset {\mathbb{P}}^d$ be a $m$-dimensional variety in $d$-dimensional projective space. Let $k$ be a positive integer such that $\left(\genfrac{}{}{0px}{}{m+k}{k}\right)\le d$. Consider the following interpolation problem: does there exist a variety $Y\subset {\mathbb{P}}^d$ of dimension $\le \left(\genfrac{}{}{0px}{}{m+k}{k}\right)-1$, with $X\subset Y$, such that the tangent space to $Y$ at a point $p\in X$ is equal to the $k$th osculating space to $X$ at $p$, for almost all points $p\in X$? In this paper we consider this question in the toric setting. We prove that if $X$ is toric, then there is a unique toric variety $Y$ solving the above interpolation problem. We identify $Y$ in the general case and we explicitly compute some of its invariants when $X$ is a toric curve.