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Some theorems are true in vector spaces or in manifolds for a given dimension $n$ but become false in higher dimensions.

Here are two examples:

  • A positive polynomial not reaching its infimum. Impossible in dimension $1$ and possible in dimension $2$ or more. See more details here.
  • A compact convex set whose set of extreme points is not closed. Impossible in dimension $2$ and possible in dimension $3$ or more. See more details here.

What are other "interesting" results falling in the same category?

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    $\begingroup$ Related: mathoverflow.net/questions/5372 $\endgroup$ – Emil Jeřábek supports Monica Sep 14 '14 at 19:12
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    $\begingroup$ This question could get more interesting if extended to fractals... $\endgroup$ – Nate Eldredge Sep 15 '14 at 4:41
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    $\begingroup$ Every cubic polynomial map from $\mathbb C^n$ to $\mathbb C^n$ with nowhere vanishing Jacobian is a bijection. This is true for $n \leq 17$ but fails for larger $n$. Just kidding... but perhaps someday I'll edit the answer, change 17 appropriately and will gain some upvotes. $\endgroup$ – Marty Sep 15 '14 at 12:55
  • $\begingroup$ Not sure it counts as "dimension", but for the sake of completeness: 1-generated groups vs 2-generated groups... $\endgroup$ – TT_ Sep 16 '14 at 22:44
  • $\begingroup$ "A positive polynomial not reaching its minimum." Better to say, not reaching its infimum, or, not having a minimum. $\endgroup$ – Gerry Myerson Mar 6 '17 at 23:18

39 Answers 39

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A set X is said to be m-convex m>=2 if for every m distinct points in X at least one of the line segments determined by those points belongs to X.

For compact sets and m=3 the decomposition into convex, aka 2-convex ,sets has been known for many years. In the plane F.A.Valentine of UCLA showed in 1957 that every 3-convex set was the union of 3 convex sets , see the 5 pointed star on the USA flag. H.G.Eggleston in 1976 gave an example of a compact 3-convex set in R4 which was not the union of finitely many convex sets. In R3 some years ago I outlined an easy proof in sci.math.research that in R3 a compact 3 convex set was the union of 4 convex sets, using the 4 colour theorem of graph theory.

For information but off topic : Many years ago much was published in the Israel Journal Of Mathematics on planar decomposition, into convex sets , bounds as a function of m for compact m-convex sets. There was also a comprehensive treatment by M. Breen of planar non-closed 3-convex sets in 1977. I know of no associated higher dimension work.

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By Liouville's theorem, a bounded subharmonic function in $\mathbb{R}^n$ is a constant. This holds not true if $n\ge 3$.

There are many similar corresponding facts in potential theory.

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$S^{2}$ and $\mathbb{R}^{2}$ satisfies the Poincare Bendixon theorem but this theorem is not satisfied by higher dimensional spheres or Euclidean spaces.

For a related MSE post see the following.

https://math.stackexchange.com/questions/861231/extension-of-poincar%C3%A9-bendixson-theorem-to-mathbbr3

As another example: For $n>8$ there is no a $n$-dimensional subvector space of $M_{n}(\mathbb{R})$ which all non zero elements are invertible matrix. The only possible $n$ are $n=1,2,4,8$. Such subvector spaces correspond to matrix representation of real numbers, complex numbers, Quaternions and Cayley numbers, respectively.

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A series of essentially equivalent statements on zonoids which hold for $d=2$ but fail for $d\geq 3$:

any convex symmetric polytope in $\mathbb{R}^d$ is a Minkowski sum of segments;

any convex symmetric body in $\mathbb{R}^d$ is a section of a unit ball in $L^1$-type space (for polytope finite-dimensional hyperoctahedron is enough);

any convex symmetric body in $\mathbb{R}^d$ is a projection of a unit ball in $L^{\infty}$-type space (for polytope finite-dimensional cube is enough);

any Banach norm in $\mathbb{R}^d$ may be expressed as $\|x\|=\int |(x,y)| d\mu(y)$ for some Borel measure $\mu$ on $\mathbb{R}^d$, where $(x,y)$ is a scalar product;

for any norm $\|\cdot\|$ on $\mathbb{R}^d$ and any vectors $v_1,u_1,\dots,v_n,u_n$ we have $\sum \|u_i\|\geq \sum \|v_i\|$ provided that $\sum |(u_i,y)|\geq \sum |(v_i,y)|$ for any vector $y$.

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If $F=(f_\lambda)_{\lambda\in\Lambda}$, is an analytic family of quadratic-like maps (with some conditions), and $M_F$ is the set of $f_\lambda$ with connected Julia set, then there is a homeomorphism of $M_F$ with the Mandelbrot set $\{c\in\mathbb{C}:f_c(z)=z^2+c\text{ has connected Julia set}\}$. See http://www.math.cornell.edu/~hubbard/PolyLikeMaps.pdf. Douady called this the "miracle of continuity" coming from the measurable Riemann mapping theorem.

Analytic families of polynomial-like maps of degree $\geq 3$ have discontinuous straighting maps. See http://arxiv.org/pdf/0903.4289v2.pdf.

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It's interesting that no one has so far mentioned Pythagorean triples and Fermat's Last Theorem :)

EDIT: With sufficient imagination any integer can be interpreted as the dimension of an appropriate vector space. However, in this particular case hardly any imagination is necessary: it is in dimensions 1 and 2 only that there exists 3 hypercubes with integer sides such that the sum of volumes of the first two is equal to the volume of the third one.

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    $\begingroup$ What does this have to do with vector spaces of increasing dimension? The Fermat curve of degree $n$ defined by $x^n + y^n = z^n$ is an algebraic plane curve. $\endgroup$ – S. Carnahan Sep 23 '14 at 2:54
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    $\begingroup$ RW means that $x^n$ is the hyper-volume of an hypercube of size $x$ in an Euclidean space of dimension $n$. Fermat Last Theorem is the statement that in dimension $n \geq 3$, the sum of the volumes of two hypercubes with side of (positive) integral length is not the volume of such an hypercupe. The same statement fails in dimensions $n = 2$ and $n=1$. $\endgroup$ – Joël Sep 24 '14 at 6:09
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The James-Stein Estimator.

Suppose $Y \in \mathbb{R}^n$ is a Gaussian vector with unknown mean $\mu$ and known spherical variance $\sigma^2I$. Given an observation of $Y$, we are interested in finding an estimator $\hat{\mu}$ of $\mu$ which minimizes the expected mean squared risk $R(\hat{\mu}) := E(||\hat{\mu} - \mu||^2)$.

For $n \le 2$, the Gauss-Markov estimator, $\hat{\mu}_{GM} = Y$ minimizes $R(\hat{\mu})$. But for $n \ge 3$ the James-Stein estimator

$$ \hat{\mu}_{JS} = \left(1 - \frac{(n-2)\sigma^2}{||Y||^2}\right)Y $$

satisfies $R(\hat{\mu}_{JS}) \le R(\hat{\mu}_{GM})$. Note however, that the James-Stein estimator itself does not minimize the mean squared risk.

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Let $B_2^n$ denote the Euclidean unit ball in $\mathbb{R}^n$. Then the Minkowski-sum $B_2^{n-1}+B_2^n$ is a zonoid whose polar is also a zonoid for $n\leq 4$, but not for $n\geq 6$. (here).

P.S -- I never did get to check the case $n=5$.

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Theorem(Samelson). The only Euclidean spheres that can be made into topological groups are $\mathbb S^0,\mathbb S^1$ and $\mathbb S^3$.

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