Locus of points fixed distance from a square

Question

Suppose I have the figure of a square like so:

If I am to sketch the locus of points one inch from the figure, then what would that look like? The outer figure would clearly be like that of another square with sort of "curved corners"; that is, the corners would be "bent" so as not to increase the distance around the corners.

Question: What would the interior figure look like? Again, it would look mostly like another square, but what exactly would the corners look like? It seems odd to me because now we are trying to add distance to corners of the figure we are coming up with as opposed to smoothing out or bending the corners for the exterior figure. What would the interior figure be?

Answer 1

I think we have to clarify the definition of the distance of a point $x$ from a figure $F$, which I would expect to be $d(x,F) = min\left\{dist(x;f); f \in F \right\}$. If we agree on this definition, then the "interior image" will be again a square, because all the pink regions and red segments have smaller distance.

Answer 2

The "parallel" square with corner triangles added $ (12345678) $ need not be smooth or continuous because the square itself is not among such figures. There are sudden jumps at corners of the square $(23,34,67,81).$