The path->difference operation could help:
to use this the yellow A needs to be a path rather than a letter. If this is not the case, convert the letter into a path. Same for the stripes. If they are strokes rather than paths, convert them.
select both the yellow letter and a white bar (red in my picture) and choose the path->difference menue
As you probably have guessed and at least one long time member has confirmed, Inkscape can be used to make also geometry illustrations. But you have no easy way to input formulas for wanted 3D shapes which you want to see rendered as 2D drawings. Only XY plane shapes can be inputted as parametric curves.
Images of 3D compositions must be drawn manually or ...
Remove the fill, if there's one, let the black outline stroke stay. Make a spare copy of your shape, you may need it later.
Select the shape. Goto Path > Stroke to Path. That transforms the outline stroke to filled area.
Use the shape to clip the image. The outline acts as a clipping mask, because it's now a filled area.
Not asked Scrap the ...
With the paths selected as in the image, click Path, Break Apart. If that doesn't split them into individual paths and objects, I'll be surprised, but you would then use the Ungroup option. Both (or more) paths will remain selected, but you'll see marquee boxes in more than one location after the break apart action.
This can be done, but note that Division can only be used with one path at a time. But that's easy to fix. Also there's no need to convert the shapes to paths beforehand.
draw a square (rectangle)
draw the four rectangles over it (or copy and paste one, three times)
select the four rectangles and do Path > Combine
select all of the rectangles, now Path > ...
I was able to accomplish the task, but my methodology isn't likely to be the most efficient. Using the "snap to midpoint" feature, I drew a line across the midpoint of each edge, then selected the square and line. Using path, division worked and created two equal squares. Repeating this created two more on the other segment remaining.
You ask for four ...
The complextity of the elements somehow seems to exceed limits of PostScript or PDF. If you save your SVG as Optimized SVG, you can reduce that complexity quite a bit.
While Inkscape still has the same problems with the optimized version, I used another app, Sketch to open it and export the PDF, and it seems to work fine.
You can't edit bitmaps in Inkscape. Inkscape is a vector image editor.
You could redraw the graphic in Inkscape however. It's not too complicated and shouldn't take too long.
All it consists of is two copies of the hypercube, one with a red stroke, one with a blue stroke on a layer above, with the blending mode set to "Darken". The blue layer is then ...
The quick way would be to use gimp to make the background of the image transparent, then it won't matter what colour your background has.
Add an alpha channel to the image:
select the colour to alpha option:
choose the background colour, white in your case
Image with transparent background:
For working with a vector image I suggest a different workflow. Instead of trying to fill the gaps, I suggest to make a copy of the whole shape, delete all the interior nodes (e.g. by using the "edit path" menu or Ctrl+Shift+K to break apart and then Ctrl++ to union [thanks at @Moini for this tip]) so that only the other edge remains. This shape can now be ...
Draw manually a black intermediate shape and blur it, There's Blur slider in the fill coloring dialog. This can be used if the exact form isn't critical and a linear gradient isn't a must.
Inkscape's extension Generate from Path > Interpolate can make automatically several intermediate versions and by interpolating the style one gets also intermediate ...
Inkscape has no background attribute as such, although it does have pagecolor which can be set in the document properties (it shows up as "background" in the UI, but appears as "pagecolor" in the XML). The problem however is that the pagecolor attribute is not supported in SVG and so it will be lost when you export as plain SVG.
Instead you could try ...
I think this would be easier to draw manually with the Pen tool in Illustrator, or the Bézier tool in Inkscape. At least that way you can get a nice smooth curve. There's nothing automated that will beat the human eye, and a little skill.
All you need is a some practice. If you have a lot of these to do, then you'll get better each time.
Seriously: No automatic tracing can extract the one and only right path under your dot chain. That's because the tracing should know the mathematical criteria for the right path. You may know it but I can only make a guess. Especially ambiquous for automatic methods are the crossings. There's zero information of should it be X or ><. Another difficulty is ...
If you have overlapping closed paths, the results will be different for Union versus Combine. If not overlapping, then there's no difference. There can also be differences with the behaviour of Combine itself, depending on the direction of paths.
I must admit analyzing SVGs properly, especially this one, is a little too complex for non-programmers. But do a workaround:
Make a bitmap copy of the problematic shape. It's in the Edit menu. Remove or close the original in the Objects panel.
You do not lose anything because it already seems to be rasterized, only wrapped in a complex way in the file ...