Name for (n-k) dimensional feature of n-dimensional polytope

Question

As per Wikipedia,

In the theory of high-dimensional convex polytopes, a facet or side of a $d$-dimensional polytope is one of its $(d − 1)$-dimensional features, a ridge is a $(d − 2)$-dimensional feature, and a peak is a $(d − 3)$-dimensional feature.

I want to know is there any general name for an $(n-k)$ dimensional feature of an $n$-dimensional convex polytope?

Answer 1

The best you can hope for is something recursive like a $k$ face or something, because $n$ is arbitrary and we can't write down arbitrarily many names!

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Answer 2

A $k$-face has dimension $k$. A face of dimension $n-k$ is an $(n-k)$-face. As pjs36 mentioned in comments, it's also called a face of codimension $k$.

If you want to introduce terminology, you might consider using "$i$-ridge" to mean a face of codimension $i+1$ (or dimension $n-i-1$). That way, an $i$-ridge is dual to an $i$-face. A facet is a 0-ridge, dual to 0-faces; ridges are 1-ridges; "peaks" are 2-ridges. The polytope itself would be a $(-1)$-ridge, and is dual to the empty face, which is considered to have dimension $-1$.

I'd be happy with that definition, but I'm not aware of anyone having used it.