Projection of an element of the $n$-simplex onto subset

Question

Let $\mathbb{S}^{n}$ denote the $n$-dimensional probability simplex and let $\{e_1,...,e_{n+1}\}$ be the canonical basis of $\mathbb{R}^{n+1}$. Consider the subset $\mathbb{S}^{n}(K) \subset \mathbb{S}^{n}$, such that $K \subset \{e_1,...,e_{n+1}\}$, defined as the vectors of $\mathbb{S}^{n}$ that can be written as linear combinations of elements of $K$. For example, if $n=2$ and $|K| = 2$, then $\mathbb{S}^{n}(K)$ is one of the edges of the $2$-simplex.

I'm trying to find a general formula for the minimum (euclidean) distance between an arbitrary element $x \in \mathbb{S}^{n}$ and some subset $\mathbb{S}^{n}(K)$. Is it already known? Is it the orthogonal projection of $x$ onto span$(K)$? How can I calculate it?

Answer 1

$\newcommand\S{\mathbb S}$Let $|\cdot|$ denote the Euclidean distance and let $k\ge1$ denote the cardinality of $K$. Then for any $x\in\S^n$ and $y\in\S_n(K)$ $$|y-x|^2=\sum_{j\notin K}x_j^2+\sum_{j\in K}(y_j-x_j)^2.$$ More specifically, let now $y$ be a minimizer of $\sum_{j\in K}(y_j-x_j)^2$ given $\sum_{j\in K}y_j=1$ (disregarding for now the condition that $y_j\ge0$ for all $j\in K$). Then, using (say) Lagrange multipliers, we see that $y_j-x_j=c$ for some real $c$ and all $j\in K$. So, $1=\sum_{j\in K}y_j=kc+\sum_{j\in K}x_j$, so that $c=\sum_{j\notin K}x_j/k\ge0$, and hence $y_j=x_j+c\ge0$ for all $j\in K$. So, the minimizer $y$ is actually in $\S^n(K)$. So, the minimum distance from $x$ to $\S^n(K)$ is $$\sqrt{\sum_{j\notin K}x_j^2+kc^2} =\sqrt{\sum_{j\notin K}x_j^2+\frac1k\,\Big(\sum_{j\notin K}x_j\Big)^2}.$$