What is the geometry of points inside a square, that are equally distant from the square

Question

Image: A visual of the problem

For the square ABCD, what would be the geometry of points that are equally distant (distance r) from all points of the square?

How would the shape written out by r look like, and is it possible to define it with a function?

EDIT: "equidistant from all points of the square" means that any point on the vertices of the square ABCD is equally distant to the shape drawn by r on the inside. A more real-life example of the problem could be seen on this image, where the square represents the seats and the problem would be "what shape should the table be, so that all people sitting on the seats (all points on the ABCD square) are equally distant to the table?". If it's a square table, it's obviously not ideal, because the person at any of the edges of the square is much further away from the table than the person in the middle of the distance AB for example.

Answer 1

There is no point equally distant to all points of a square. Indeed, if $P$ were such a point, then the triangle $PAB$ is isoscele, as $PA=PB$. Therefore, the orthogonal projection of $P$ on the line $(A,B)$ (sorry if the notation is unusual) is the middle point of the segment $[A,B]$, call it $I$. Such a point belongs to the square, thus $PI=PA=PB$, however, $PAI$ is a square triangle, thus $PA>PI$, contradiction.

Answer 2

If the square $ABCD$ has dimension $s,$ then the locus you want is a smaller square contained in the finite region defined by the first, and concentric with it, so that it has dimension $s-2r.$