References to a Convex Analysis result.

Question

I am writing an article for future publication. Due to the limit of pages that the publication delimits for your published articles I do not have space to include the demonstration of several secondary results. One of these secondary results is the result below that I found only in a single references.

Let $K$ a convex and compact subset of $\mathbb{R}^n$. Then

1. for all $x\in\mathbb{R}^n$ there is one, and only one, $x_K\in K$ such that $\|x-x_K\|\leq \| x-c\|$ for all $c\in K$;

2. the map $\mathbb{R}^n\ni x\mapsto x_K\in K$ is uniformly continuous and for all $x,y\in\mathbb{R}^n$ we have $\|x_K-y_K\|\leq \|x-y\|$.

But there are two problems with this reference.

• First, the reference is a somewhat inaccessible book, because it is a book written in Portuguese and the last edition is from 1976. That is, the book is no longer editable

• Second, the result is in the pre-hilbertian space context where $ K $ has the property of complete rather than the property of being compact.

My question. Is there any reference to the above result that is accessible and restricted to finite-dimensional spaces?