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Generalize the best lower bound for c(A1 + A2) to higher dimensions

Establish a generalization of Theorem \ref{Lower_bound} to arbitrary non-empty compact sets A1 and A2 in R^n for n ≥ 2 by determining a function F_n: [0, n]^2 → R such that F_n(c1, c2) equals the minimum possible Schneider non-convexity index c(A1 + A2) over all pairs with fixed indices c(A1) = c1 and c(A2) = c2. This will extend the one-dimensional optimal lower bound to higher dimensions and enable characterization of the Schneider region S_n(2).

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Background

The paper defines the Schneider region S_n(m) via the Schneider non-convexity index c and completely characterizes S_1(2). A key ingredient is Theorem \ref{Lower_bound}, which provides the optimal lower bound for c(A1 + A2) in one dimension as a function of c(A1) and c(A2).

To extend this program to higher dimensions and characterize S_n(2), the authors note that a generalization of the one-dimensional lower bound to Rn is needed. Without such a generalization, only partial characterizations (e.g., Proposition \ref{partial_char}) are available.

References

We note here that, while in general finding a complete characterization of the region S_n(m) for any n,m looks extremely complicated, it would be interesting to achieve a characterization of the region S_n(2), for n≥2. The main difficulty that has prevented us from achieving this is that we do not know the generalization of Theorem \ref{Lower_bound} for arbitrary compact sets in \mathbb{R}n.

— Measuring the convexity of compact sumsets with the Schneider non-convexity index (2405.00221 - Meyer, 30 Apr 2024) in Remark following Theorem \ref{char_thm}, Section “The Schneider region”, Subsubsection “Characterization for two sets in one-dimension”