Squeeze the center of an object?

[tlujan0001R7HAG]

I have a group of tiles that I'd like to  'squeeze' in the center.  I've tried to create sort of an inverse fillet using the pipe tool, but it only works on the edge of the object, not the middle. Here is an example of the 'squeeze' shape I'm trying to create (it does not have to be the exact same angle pictured here).  

My shapes were imported via dxf from Inkscape and I extruded them from a sketch. In Inkscape the paths are continuous. They look like this:

Can someone get me started in the right direction?

[g-andresen]

Hi,

Is it about such a groove?

1. create a midplane

2. project profiles into a sketch 

3. create pipes

günther

[tlujan0001R7HAG]

@g-andresen Thank you! A very elegant solution and the video was easy to follow. I really appreciate it and will give this a go in the morning.

[tlujan0001R7HAG]

I've attempted the solution you suggested and can't quite get this to work.  The pipe profile needs to sit deeper into the face of the tile (1mm from the profile edge toward the inside of the tile). When trying to use the pipe command I'm getting an error, "Error: The given path is not tangent continuous around tight corners. To make self-intersecting sweep work, the circular path segments should be tangent continuous to the adjacent path segments."

I've tried to use the sweep command to create a rectangular profile shape but am not sure how to do this for each tile.