Meckes - A direct proof for the positive definiteness of four point metric spaces

Abstract

The positive definiteness is a property of a metric space which ensures nice behaviour of its magnitude. For any four-point metric space, the positive definiteness was first established by Meckes, where an embedding theorem is invoked. The aim of my talk is to explain a direct proof for this fact without invoking an embedding theorem. I also discuss a possible condition for the magnitude of a finite metric space to obey the inclusion-exclusion principle with respect to a specific choice of subspaces. This condition is suggested by the direct proof, and its validity is verified when the number of points is small.

https://case.edu/artsci/math/mwmeckes/

Notations

• $A$ compact metric space
• $tA$ skalierter Raum
• $M(A)$ all signed Borel measures auf $A$. (i.e. negative values are allowed)
• $P(A)$ is the set of Borel probability measures.

Average distance $Z\mu$

For a signed measure $\mu$ and $x\in A$ define a kind of average distance: $Z\mu (x)={\int }_X{e}^{-d(x,y)}d\mu (y)$

Magnitude

Ursprünglich invertieren wir eine Matrix, die die Abstände zur allen anderen Punkten enthält. Was genau ist hier die Verallgemeinerung? Im Kontinuierlichen ist so etwas natürlich nicht möglich. However there are three ways of making this work:

Definition Magnitude metric space

1. If we assume that there is a $\mu$ such that $Z\mu$ is everywhere $1$ (a weight measure), then the following definition makes sense: $Mag(A)=\mu (A)=\int Z\mu d\mu$
2. Assuming $A$ is of Metrischer Raum von negativem Typ. Then $\text{Mag}(A)={\sup }_{\mu \in M(A)}\{\frac{\mu (A{)}^2}{\int Z\mu d\mu }:\mu \in M(A)\}$
3. Again assuming negative type there is also $Mag(A)=\inf \{\Vert f{\Vert }_H^2:f=1\}$ where $f$ are all functions that have constant values over $A$.

Meckes showed that all variants are equivalent if defined.

Maximum diversity

We define a maximum diversity for metric spaces. This measures how many different points a space effectively contains. Isn’t this the same as magnitude?.

Maximum entropy

The maximum entropy was defined by Roff and Leinster as: $\text{Div}(A)=\sup \mu \in P(A){\left({\int }_X(Z\mu {)}^{q-1}d\mu \right)}^{\frac{1}{1-q}}$ where $q\in [0,\infty ]$, though it turns out to be independent of $q$.

Basic inequalities

Meckes collected some inequalities about diversity and magnitude in metric spaces:

Basic Inequalities

• For $A\subseteq B$: $1\le Div(A)\le Div(B)<e^{diam(B)}$.
• For $A\subseteq B$, $B$ of negative type, then $1\le Mag(A)\le Mag(B)$.
• Fundamental inequality: if $A$ of negative type then $Div(A)\le Mag(A)$.
• Magnitude is lower semicontinous on spaces o negative type. If $A_k\to A$ (Convergence in Gromov-Hausdorff) then $Mag(A)\le {\mathrm{liminf}}_{k\to \infty }Mg(A_k)$ This can be a strict inequality. Connect seperate 3 Punkte complete with a complete $3$-Graph. Then the magnitude converges to $6/5$, i.e. more then the limit $1$.

Lower Bound Diversity

• It’s easy to show that $\text{Div}(A)\ge \frac{\text{vol}(A)}{n!\text{vol}(B)}$ Where $A$ is a subset of an $n$-nim normed space whose unit ball is called $B$.
• The packing number $P(A,ϵ)$ is a notion of the effective number of points at a particular scale. The largest number points at distance $\ge ϵ$ fro each other. Then $\text{Div}(A)\ge \frac{P(A,ϵ)}{1+P(A,ϵ)e^{-ϵ}}$

Slogan: The magnitude measures the effective number of points. Isn’t this something the Diversity does?

Upper bound Diversity

• Aishwarya: Let $A\bigcup A_i$. Then $Div(A)\le \sum Div{A}_i$
• The covering number ist the smalles cardinality of an $ϵ$-net. The previous bullet points implies $\text{Div}(A)\le N(A,ϵ)e^{2ϵ}$
• Packing and covering numbers are basically the same: $P(A,2ϵ)\le N(A,ϵ)\le P(A,ϵ)$. This makes the entropy related to the minkowski dimension ${\dim }_{mink}(A)={\lim }_{e\to 0}\frac{\log N(A,ϵ)}{ϵ}$

This last piece where the diversity can be described by the packing and covering number might connect to the topological entropy of a function.

Lower Bounds for magnitude

• $Mag(B)\ge Mag(A)$ for $A\subseteq B$ with specific simple $A$.
• $Mag(A)\ge Div(A)$ (gives us lower bounds on magnitude)

Upper bounds for magnitude

• The definition of the magnitude as an infimum (see above) gives an immediate bound by a norm of a function. Sadly the Sobolev-type norm is very hard to calculate and only useful in specific circumstances. One such example is a subset $A$ of euclidean space. There we have $\text{Div}(A)\le \text{Mag}(A)\le c_n\text{Div}(A)$ where $c_n$ is a dimensional constant.

Best method to get upper bound:

• Approximate $A$ by spaces with explicit weight measures
• Use semicontinuity $Mag(A)\le {\lim }_{k\to \infty }Mag(A_k)$

Finiteness

Let $E$ be a finite dimensional space of negative type

One Point Property

If $A\subseteq E$ ist compact than $Mag(tA)<\infty$ for all $t>0$ and ${\lim }_{t\to 0}Mag(tA)=1$.

Sudakov's minoration inequality

If $K\subseteq l_2^n$ ist compakt and convex, then $V_1(K)\ge C{\sup }_{ϵ\ge 0}ϵ\sqrt{\log N(K,ϵ)}$

Reisner 86

$B$ is the unit ball in an n-dim normed space $E$ of negtive type.. Then $vol(B)vol(B^{\circ })\ge \frac{4^n}{n!}$ $B^{\circ }$ is the Polar set of $B$

Brunn-Minkowski-type inequalities

Aishwarya li Madiman-M 23: For $A,B\subseteq \mathbb{R}$ and $0<\lambda <1$, there is $\text{Div}((1-\lambda )A+\lambda B)\ge \text{Div}(A/\sqrt{2}{)}^{1-\lambda }\text{Div}(B/\sqrt{2}{)}^{\lambda }$

Open problems

• If $t>1$ ist $Mag(tA)\ge Mag(A)$ ist $Mag(A)\le \#A$
• It $t<1$ ist $Mag(tA)\ge Mag(A{)}^t$
• is magnitude continuous on compact convex sets in a normed space o negative type?
• Is $\frac{d}{dt}mag(tK)=\frac{{\mu }_1^E(K)}{4}$ for $K$ compakt in $E\subseteq L_1$

Underexplored avenues

• Exact values of diversity
• Concrete upper bounds for Div
• Lower Bounds on Mag better than Div
• Can we get Inequalities from magnitude homology?