Smooth centro-affine Minkowski problem and affine distance

Determine whether a prescribed affine distance can be realized in the smooth centro-affine Minkowski problem corresponding to the critical parameter p=-n.

Background

The critical case p=-n is identified as the centro-affine Minkowski problem. In the smooth setting, solving this problem is equivalent to prescribing affine distance, a longstanding issue in affine differential geometry. The paper discusses several partial results but leaves this smooth problem unresolved.

References

In the smooth setting, it amounts to prescribing the affine distance, a longstanding open problem in affine differential geometry.

— The Discrete $L_p$ Minkowski Problem for Negative $p$  (2609.01429 - Shan, 1 Sep 2026) in Section 1, Introduction