Existence and homeomorphic classification of $(m,n)$-equidistant metric continua (Popvassilev's Problem 5.2 and Conjecture 5.1)

For integers $m\ge 2$, $n\ge 1$, a metric space $(X,d)$ is called $(m,n)$-equidistant if every $m$-point subset $A\subseteq X$ has exactly $n$ points of $X$ equidistant from all of its points. Popvassilev (2008) posed the problem of determining for which pairs $(m,n)$ there exists a metric continuum — a compact connected metric space — that is $(m,n)$-equidistant, and of classifying all such continua up to homeomorphism (Problem 5.2), together with the central special case (Conjecture 5.1): every $(3,2)$-equidistant continuum is homeomorphic to the $2$-sphere, a higher-point analogue of the classical Double Midset Conjecture for circles. Only three realizations are known: the interval for $(2,1)$, the circle for $(2,2)$, and the $2$-sphere with the chord metric for $(3,2)$; the conjecture is open even for continua lying in $\mathbb{R}^3$ with the usual metric. The problem remains open.

Background

For two distinct points $a,b$ of a metric space $(X,d)$, the midset (or bisector) is the set $E(\{a,b\})=\{x\in X: d(x,a)=d(x,b)\}$. Popvassilev generalized this as follows: for a subset $A$ with at least two points, the equidistance set is $E(A)=\{x\in X: d(x,a)=d(x,b)$ for all $a,b\in A\}$, and $X$ is $(m,n)$-equidistant (for integers $m\ge 2$, $n\ge 1$) if $X$ has at least $m$ points and every $m$-point subset $A\subseteq X$ has $|E(A)|=n$. The case $m=2$ is classical: it is the $n$-point midset property ($n$-MP), with $1$-MP, $2$-MP, $3$-MP called UMP, DMP, and TMP (unique, double, triple midset property). The real line is $(2,1)$-equidistant (each pair has one midpoint), a circle is $(2,2)$-equidistant, and a continuum means a compact connected metric space. Berard (1971) proved that a connected metric space with the UMP is homeomorphic to an interval in $\mathbb{R}$; Loveland and Valentine (1978) showed that generalized midset properties characterize geodesic circles and intervals; and Loveland and Wayment (1974) posed the Double Midset Conjecture (DMC): a continuum with the DMP, i.e. a $(2,2)$-equidistant continuum, is homeomorphic to a circle. Loveland (1991) proved the DMC for continua contained in the plane with the usual metric, and his TMP results (1990, 1992) show that no continuum in the plane has the triple midset property; for arbitrary continua the DMC itself remains unresolved. Popvassilev (2008) initiated the general $(m,n)$ theory. Among his results for concrete spaces: for $k\ge 2$ the Euclidean space $\mathbb{R}^k$ contains an $(m,1)$-equidistant set if and only if $k\ge m$; any $(m,n)$-equidistant subset of $\mathbb{R}^k$ with $n\ge 1$ has all subsets of at most $m$ points affinely independent, forcing $m\le k+1$; $\mathbb{R}^k$ contains no $(m,n)$-equidistant set when $m\ge\max\{k,3\}$ and $n\ge 3$; the $2$-sphere $\mathbb{S}^2\subseteq\mathbb{R}^3$ with the chord metric (the Euclidean distance of $\mathbb{R}^3$ restricted to the sphere) is $(3,2)$-equidistant, since any three points of $\mathbb{S}^2$ lie on a unique circle whose pole pair is exactly the equidistance set; but neither $\mathbb{S}^3$ nor $\mathbb{R}^4$ contains any $(4,2)$-equidistant set; and the real projective plane $\mathbb{P}^2$ (the antipodal quotient of $\mathbb{S}^2$ with the induced metric) contains a $6$-point set with all pairwise distances equal, hence $(m,n)$-equidistant subsets whenever $m+n\le 6$. These discrete and spherical results delimit but do not touch the continuum question, since continua are connected (and uncountable), while the extremal discrete examples — finite sets with a constant metric — are disconnected. Note also that a $(3,2)$-equidistant space need not have the DMP: on $\mathbb{S}^2$ with the chord metric the midset of two points is a great circle, so Conjecture 5.1 is a genuinely new higher-point analogue of the DMC rather than an instance of it. At the end of the paper Popvassilev states Conjecture 5.1 — every $(3,2)$-equidistant continuum is homeomorphic to $\mathbb{S}^2$ — and Problem 5.2, which asks for the full existence and classification picture, admitting he cannot prove the conjecture even for continua contained in $\mathbb{R}^3$ with the usual metric.

Problem Statement

Determine all pairs of integers $(m,n)$ with $m\ge 2$ and $n\ge 1$ for which there exists a metric continuum $X$ that is $(m,n)$-equidistant — i.e. a compact connected metric space $(X,d)$ with at least $m$ points such that every $m$-point subset $A\subseteq X$ has an equidistance set $E(A)=\{x\in X: d(x,a)=d(x,b)\ \text{for all}\ a,b\in A\}$ of exactly $n$ points — and, for each such pair, classify all $(m,n)$-equidistant continua up to homeomorphism (Popvassilev's Problem 5.2). The designated central open special case is Popvassilev's Conjecture 5.1: every $(3,2)$-equidistant continuum is homeomorphic to the $2$-sphere $\mathbb{S}^2$; this is open even when $X$ is assumed to be a subset of $\mathbb{R}^3$ with the usual Euclidean metric. The known realizations are the interval $(2,1)$, the circle $(2,2)$, and $\mathbb{S}^2$ with the chord metric $(3,2)$; whether any pair with $m\ge 4$, $n\ge 3$, or $(m,n)=(3,1)$ admits an $(m,n)$-equidistant continuum is unknown.

References

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