## mg.metric geometry – Inscribed square and convexity

Let $$Fr(X)$$ be the frontiere of any $$X$$ subset of the plane.

Does there exists $$A,B$$ convex compact sets of the plane, such that $$C:=Asetminus B$$ is connex and not empty, and such that there is no square inscribed $$Fr(C)$$ that has a vertex in $$Acap Fr(C)$$ ?

If the answer is no, this would kind of mean that the Toeplitz conjecture is kind of linked to convexity (one of the firts intersting resolved case was the case of the frontiere of a convex)

If the answer is yes… then it suggests a “fractal” way to built some counterexample.

Do not hesitate to say your “pronostic” in the comments, even if it is not argued so far, I have absolutally no intuition myself about the pronostic (I would say it must be “YES” but I would be influenced by the fact that according to me, if it is “YES”, then there is a counterexample not far, and this would have been found – but I might as well underevaluate the difficulty of “YES=> counterexample” …)

## mg.metric geometry – Collinearity in tangential pentagon

I am looking for a proof of the following claim:

Given tangential pentagon. Touching point of the incircle and the side of the pentagon,the vertex opposite to that side and the intersection point of diagonals drawn from endpoints of that same side are collinear.

The GeoGebra applet that demonstrates this claim can be found here.

## mg.metric geometry – Do Compact Universal Covers have Concentration of Measure phenomenon?

I have a sequence of compact Riemannian manifolds $$M_n$$ with $$diam (M_n) to 0$$ and finite fundamental groups $$pi_1 (M_n)$$ so that their universal covers $$L_n$$ are compact. Suppose $$diam ( L_n ) = 1$$ for all $$n$$, and their volumes are normalized so that $$vol (L_n)=1$$.

Is it true that the family of metric measure spaces $$L_n$$ satisfies some concentration of measure phenomenon? Are they a Levy family? This is closely related to this other question: Diameter of universal cover .

## mg.metric geometry – Holder-continuous barycenter maps

Let $$(X,d)$$ be a complete locally-compact metric space. We define the $$p$$-barycenter map as a continuous function:
$$beta:mathcal{P}_p(X)rightarrow X,$$
which is a right-inverse of the map associating to any $$x in X$$ its point-mass $$delta_xin mathcal{P}_p(X)$$.

It is well-known that, if $$X$$ is CAT(0) then $$beta$$ can be assumed to be $$L$$-Lipschitz for some $$Lin (0,1)$$. My question is, are there known (resp. classes of) examples of metric spaces for which $$beta$$ is Hölder continuous but not Lipschitz continuous?

## mg.metric geometry – Reference to equivariant Gromov-Hausdorff convergence

I am looking for a reference to the following notions and facts below which, I think, I can prove, but which might be known to experts.

Let us fix a finite group $$G$$. Consider the class of all compact metric spaces equipped with an action of $$G$$ by isometries. The elements of it are triples $$(X,d,a)$$ where $$(X,d)$$ is a compact metric space, $$a$$ is an action.
Definition. The $$G$$-equivarint GH-distance between two such triples is $$d^G_{GH}((X,d,a),(Y,h,b)):= inf_{D} D_H(X,Y),$$
where the inf is taken over all $$G$$-invariant metrics $$D$$ on the disjoint union of $$X$$ and $$Y$$ extending the original metrics $$d$$ and $$h$$ on them; $$D_H$$ denotes the Hausdorff distance.

.Proposition
(1) $$d^G_{GH}$$ satisfies the triangle inequality.

(2) $$d^G_{GH}((X,d,a),(Y,h,b))=0$$ if and only if there exists a $$G$$-equivariant isometry between $$X$$ and $$Y$$.

Let us denote by $$mathcal{X}^G$$ the set of isomorphism classes of such triples. Thus it is the metric space with the induced metric.

Let $$mathcal{X}$$ denote the usual space of isometry classes of compact metric spaces equipped with the usual (non-equivariant) Gromov-Hausdorff metric.

Proposition.
The canonical map (forgetting the action)
$$mathcal{X}^Gto mathcal{X}$$
is proper.

## mg.metric geometry – Simplex invariants?

Let $$k=s$$ be a positive definite symmetric function and a simililarity over the natural numbers such that $$k(a,a) = 1$$ for all natural numbers $$a$$.

A similarity $$s:Xtimes X rightarrow mathbb{R}$$ is defined in Encyclopedia of Distances as:

1. $$s(x,y) ge 0 forall x,y in X$$

2. $$s(x,y) = s(y,x) forall x,y in X$$

3. $$s(x,y) le s(x,x) forall x,x in X$$

4. $$s(x,y) = s(x,x) iff x=y$$

We can define the metric $$d(x,y) = sqrt{k(x,x)+k(y,y)-2k(x,y)}=sqrt{2(1-k(x,y)}$$.

Every $$3$$ point metric space can be embedded in $$mathbb{R}^2$$ as a triangle, (See https://math.stackexchange.com/questions/3393140/whats-the-name-of-this-surface-a2b2c22abc-1-0 ), and hence for each triple of distinct natural numbers $$x,y,z$$ we get a triangle.

For three points $$x,y,z$$ in a metric space, we can define (using the law of cosines) the following quantity:

$$S(x,y,z) = frac{d(x,y)^2+d(y,z)^2-d(x,z)^2}{2d(x,y)d(y,z)}$$

We then have:

$$pi = arccos(S(x,y,z))+arccos(S(z,x,y))+arccos(S(y,z,x))$$

wich I consider to be an invariant, because it does not depend on the metric space $$(X,d)$$.
(and has a nice effect of yielding some amusing formulas for $$pi$$ https://math.stackexchange.com/questions/4183066/some-formulas-for-pi ).

Now my naive question is, if it is possible to define other invariants for each simplex corresponding to a subset of $$n$$ natural numbers, which is independent on the function $$k=s$$ chosen.

The only thing we know are the properties above about $$k=s$$ or $$d$$.

(We can also assume that the determinant of the Gramian matrix $$k(x_i,x_j)$$ for each subset $$x_i$$ of pairwise distinct $$x_i,x_j$$ is not zero.)

## mg.metric geometry – The product of the lengths of two line segments that belong to Newton line

I am looking for the proof of the following claim:

Consider a family of bicentric quadrilaterals with the same inradius length. Denote by $$P$$ and $$Q$$ the midpoints of the diagonals, and by $$I$$ the incenter. Then, $$|PI| cdot |QI|$$ has the same value for all quadrilaterals in the family.

The GeoGebra applet that demonstrates this claim can be found here.

## mg.metric geometry – A circle with radius R, find the total area of all the circles added together

A circle with radius R is shown below in figure a. In figure b, two circles with a radius of 1/2 R are placed on top of the original circle from figure a. In figure c, four circles with a radius of 1/4 R are placed on top of the circles from figure b.

The image

Assuming this pattern continues indefinitely, find the total area of all the circles added together.

(With steps would be wonderful, I really don’t understand)

## mg.metric geometry – Does codimension-1 collapsing with bounded curvature have boundary?

Let $$(M^n,g_i)$$ be a sequence of smooth complete Riemannian manifold with $$|sec_{g_i}| le 1$$. Suppose $$(M_i^n,g_i)$$ converges to a limit space $$(X^{n-1},d)$$ in the Gromov-Hausdorff sense, where the Hausdoff dimension of $$X$$ is $$n-1$$.

Can we show that $$X$$ contains no boundary point? Here, a point is a boundary point of $$X$$ if its tangent cone is isometric to $$mathbb R^{n-2} times mathbb R_+$$.

## mg.metric geometry – Cutting Convex Regions into equal diameter and equal least width pieces – 3

We add a bit to Cutting convex regions into equal diameter and equal least width pieces – 2. There, we asked, for example: If we divide a 2D convex region C into n convex pieces such that the maximum diameter among the pieces is a minimum, will we necessarily also get all pieces with same diameter?

Questions:

1. If the maximum diameter among n convex pieces into which C is being divided is to be a minimum, will it automatically guarantee a minimum of the average diameter among pieces?

(If minimum diameter among pieces is to be maximized, we have degenerate pieces, although one can also assert that average diameter is then maximized).

One can also ask in the same spirit:

1. If the maximum perimeter among n pieces is to be minimized, will it have any automatic implication on the average perimeter among pieces?
2. If the maximum (minimum) least width among n pieces from C is to be minimized (maximized), will it have any impact on the average least width (The least width of a region is the least distance between any pair of parallel lines that touch the region)?