Define the Euler measure of a polyhedral set as the Euler integral of its indicator function. It is easy to show by induction that the
Euler measure of a closed bounded convex polyhedron
is always 1 (independent of dimension), while the Euler measure of a 
-D relative-open bounded convex polyhedron
is 
.
Euler Measure
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References
Propp, J. "proof of Euler-Poincare formula." math-fun@cs.arizona.edu posting, Aug. 30, 1996.Referenced on Wolfram|Alpha
Euler MeasureCite this as:
Weisstein, Eric W. "Euler Measure." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EulerMeasure.html