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Suppose I have the figure of a square like so:

enter image description here

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?

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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.enter image description here

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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 $(23,34,67,81).$

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    Parallel square? And what is your notation for the corners of the other square you mention? I find that rather confusing. Also, I have seen in certain places that the locus for the inside of a rectangle is simply another rectangle with sharp edges, but how is that so? The distance between corners is clearly greater than the distance between edges. – 2017-02-19
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    Also what do you mean by "the square itself is not among such figures." Among what figures? What in your figure is supposed to be the "interior" locus? The lighter shaded lines? I'm having a hard time understanding your answer. – 2017-02-19
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    The square is not smooth at its corners it cannot be counted one among closed smooth curves.. So a parallel profile built up by vector normal tips cannot be smooth continuous.If you say it is a smaller square what about parallel points corresponding to inner tiny rectangles like $23?$ – 2017-02-19