Large convexly independent subsets of Minkowski sums

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Problem: Swanepoel--Valtr Conjecture 7 is formalized as follows. A finite set $X\subset \mathbb R^3$ is convexly independent if every point of $X$ is an extreme point of $\operatorname{conv}X$. The conjecture asserts that there exists $\varepsilon>0$ such that for all four two-point sets $A_i=\{p_i,q_i\}\subset\mathbb R^3$ with $\|p_i-q_i\|_2<\varepsilon$, the set

is not convexly independent. This is exactly the statement quoted from Conjecture 7; the ordered union is the same as the union over $i<j$.

Result: The conjecture is false. In fact, for every $\varepsilon>0$ there is such a configuration whose 24 cross-midpoints are convexly independent.

Let $a_i^0,a_i^1\in\mathbb R^3$ be

For $i<j$, $\alpha,\beta\in\{0,1\}$, write

The following table gives, for each $x_{i\alpha,j\beta}$, an exposing vector $u$. The listed positive integer $m$ is the minimum of the ten scalar products

where $k\notin\{i,j\}$ and $\gamma\in\{0,1\}$.

Because every listed $m$ is positive, for each $x_{i\alpha,j\beta}$ the corresponding $u$ strictly separates it from every other $x_{r\rho,s\sigma}$ with $r<s$. Hence all 24 points $x_{i\alpha,j\beta}$ are vertices of their convex hull.

Now let

Given any $\varepsilon>0$, choose $0<\lambda<\varepsilon/D$ and set

Then $\| \lambda a_i^0-\lambda a_i^1\|_2<\varepsilon$, while the cross-midpoint set is just the homothetic image by factor $\lambda/2$ of the convexly independent set above. Homotheties preserve convex independence. Thus the conjectured $\varepsilon$ cannot exist.

Verification audit: the constructed $A_i$ are genuine two-point subsets of $\mathbb R^3$; the union over ordered $i\ne j$ equals the 24-point union over $i<j$; each point has an explicit strict exposing functional; scaling makes the within-pair distances arbitrarily small without changing convex independence.

Citation: No external resolution is used here. The conjecture is from Swanepoel and Valtr, “Large convexly independent subsets of Minkowski sums,” Electron. J. Combin. 17 (2010), R146, Conjecture 7.