Abstract
This corpus has been writing $\mathrm{Vol}(S^{d-1})$ since Paper 01. It has never once remarked that the ball this solid angle measures vanishes as the dimension grows. This paper opens that — and finds that the phrase “curse of dimensionality” covers several things with different causes, of which the sharpest is not a curse at all. NO NEW MATHEMATICAL THEOREM IS CLAIMED.
First, the volume of the unit ball peaks at $d=5$ and falls thereafter, reaching $2.37\times10^{-40}$ at $d=100$ (Section 2).
Second, where the volume went is two separate statements. Within one body the mass gathers in a thin skin ($0.99996$ of it within $0.01$ of the surface at $d=1000$); between two bodies the ball loses to the cube outright ($1.87\times10^{-70}$ at $d=100$) (Section 3).
Third, two random directions are nearly orthogonal. The mean absolute cosine is $0.0252$ at $d=1000$, matching $\sqrt{2/(\pi d)}=0.0252$ (Section 4).
Fourth, this is what people mean by the curse. The relative gap between the nearest and the farthest point collapses to $0.113$ at $d=1000$, so “nearest” stops selecting anything (Section 5).
Fifth, and this is the core. Read the other way, that same concentration is the blessing. Projecting $300$ points from $1000$ dimensions down to $100$ at random moves all $44850$ pairwise distances by $5.5\%$ on average — the curse and the blessing are the same theorem, and what separates them is only whether you wanted the function to vary (Section 6).
Sixth, the fence is what the concentration is around. The maximum of $d$ standard gaussians does concentrate, but around a value that keeps growing (Section 7).
Scope (what is not claimed)
NO NEW MATHEMATICAL THEOREM IS CLAIMED — the volume of the ball, concentration of measure, the Johnson–Lindenstrauss lemma, and the degeneration of nearest neighbours in high dimension are all standard. No machine-learning method is evaluated. On Papers 94 and 114: 94 split the word dimension into topological, fractal and special integer dimensions; 114 sorted objects by whether they know their own dimension. This paper asks what happens when the dimension is large — a different question from either. On Paper 139: that paper also mentions the curse once, but about the convergence rate of empirical measures, which is a different quantity from the geometry here. What is added is separating five things by what each is a statement about, setting the curse and the blessing side by side in numbers as two readings of one concentration, measuring the fence that concentration is around the mean and not around a constant, and writing down where the $\mathrm{Vol}(S^{d-1})$ this corpus keeps carrying actually goes.
1. Notation and position
Write $V_d=\pi^{d/2}/\Gamma(d/2+1)$ for the volume of the unit ball in $d$ dimensions and $A_d=2\pi^{d/2}/\Gamma(d/2)$ for the area of the unit sphere $S^{d-1}$. Everything sampled was sampled for this paper, with the sample size stated.
Position: From the $n=d-1$ uniqueness of Paper 01 through the exponent formula of Paper 116 to the solid angles of Paper 36, this corpus has written $\mathrm{Vol}(S^{d-1})$ over and over. What that value does as $d$ grows has never been asked.
2. Where the volume this corpus carries actually goes
Verification 2.1
| $d$ | $A_d=\mathrm{Vol}(S^{d-1})$ | $V_d$ (volume of the unit ball) |
|---|---|---|
| $3$ | $12.56637061$ | $4.188790205$ |
| $5$ | $26.31894507$ | $\mathbf{5.263789014}$ |
| $6$ | $31.00627668$ | $5.16771278$ |
| $7$ | $\mathbf{33.07336179}$ | $4.72476597$ |
| $10$ | $25.5016404$ | $2.55016404$ |
| $20$ | $0.5161378278$ | $0.02580689139$ |
| $50$ | $8.651096229\times10^{-12}$ | $1.730219246\times10^{-13}$ |
| $100$ | $2.368202102\times10^{-38}$ | $\mathbf{2.368202102\times10^{-40}}$ |
(QED)
Observation 2.2 The ball's volume peaks at $d=5$ and falls from there, while the solid angle peaks at $d=7$ — even the two peaks sit in different places. And by $d=100$ the $\mathrm{Vol}(S^{d-1})$ this corpus keeps writing is down in the $10^{-38}$ range.
3. Where the volume went is two separate statements
Verification 3.1: The fraction of a ball's volume lying within $\varepsilon$ of its surface, which is exactly $1-(1-\varepsilon)^d$.
| $d$ | $\varepsilon=0.1$ | $\varepsilon=0.01$ | $\varepsilon=0.001$ |
|---|---|---|---|
| $2$ | $0.19$ | $0.0199$ | $0.001999$ |
| $10$ | $0.65132156$ | $0.095617925$ | $0.0099551198$ |
| $100$ | $0.99997344$ | $0.63396766$ | $0.095207853$ |
| $1000$ | $1.0$ | $\mathbf{0.99995683}$ | $0.63230458$ |
| $10000$ | $1.0$ | $1.0$ | $0.99995483$ |
(QED)
Verification 3.2: The ball as a fraction of the cube of side $2$.
| $d$ | $V_d/2^d$ |
|---|---|
| $2$ | $0.78539816$ |
| $5$ | $0.16449341$ |
| $10$ | $0.0024903946$ |
| $20$ | $2.461137\times10^{-8}$ |
| $100$ | $\mathbf{1.8681821\times10^{-70}}$ |
(QED)
Observation 3.3 These are different statements. The first says where the mass of one body sits and follows from one line of algebra once $\varepsilon$ is fixed. The second compares two bodies and follows from the ball having no corners. Calling both the curse of dimensionality hides the difference.
4. Two random directions are nearly orthogonal
Verification 4.1: In each dimension take $20000$ pairs of random unit vectors and measure the absolute cosine of the angle between them.
| $d$ | mean $\lvert\cos\theta\rvert$ | $\sqrt{2/(\pi d)}$ | largest over $20000$ pairs |
|---|---|---|---|
| $3$ | $0.49988973$ | $0.46065887$ | $0.99998180$ |
| $10$ | $0.25722537$ | $0.25231325$ | $0.93703595$ |
| $100$ | $0.07953272$ | $0.07978846$ | $0.38790477$ |
| $1000$ | $\mathbf{0.02521252}$ | $\mathbf{0.02523133}$ | $0.14522426$ |
(QED)
Observation 4.2 The typical cosine falls like $1/\sqrt{d}$. In a thousand dimensions two vectors drawn at random are orthogonal for all practical purposes. This will return later on the side of the blessing.
5. What people mean by the curse
Verification 5.1: Take $2000$ points uniformly in the unit cube and look at their distances from a random query point.
| $d$ | nearest | farthest | relative gap |
|---|---|---|---|
| $1$ | $0.000287$ | $0.759371$ | $2644.59$ |
| $5$ | $0.239713$ | $1.440486$ | $5.009207$ |
| $20$ | $0.940906$ | $2.617093$ | $1.781461$ |
| $100$ | $3.304831$ | $4.780400$ | $\mathbf{0.446489}$ |
| $1000$ | $12.168306$ | $13.539749$ | $\mathbf{0.112706}$ |
(QED)
Observation 5.2 The nearest and the farthest point are at almost the same distance. The operation “take the nearest” stops selecting anything — this is what the curse actually is.
6. The core: read the other way, the same concentration is the blessing
Observation 6.1 What flattened the distances in Section 5 is the fact that a Lipschitz function of many independent coordinates is nearly constant. But that same fact also means distances survive a projection.
Verification 6.2: Take $300$ points in $1000$ dimensions ($44850$ pairs), project to $k$ dimensions with a random matrix, and measure the ratio of every pairwise distance.
| $k$ | worst distortion | mean distortion | compression |
|---|---|---|---|
| $20$ | $0.624735$ | $0.119932$ | $50\times$ |
| $50$ | $0.457538$ | $0.078553$ | $20\times$ |
| $100$ | $0.300259$ | $\mathbf{0.055155}$ | $10\times$ |
| $200$ | $0.194477$ | $0.038989$ | $5\times$ |
| $400$ | $0.160552$ | $0.029956$ | $2.5\times$ |
(QED)
Observation 6.3 A thousand dimensions crushed into a hundred, and $44850$ distances move by $5.5\%$ on average. What makes this possible is the same concentration that destroyed the nearest neighbour in Section 5. The curse and the blessing are one theorem, and what separates them is only whether the function was wanted to vary.
Observation 6.4 If the aim is to pick a neighbour by distance, not varying is a disaster. If the aim is to compress while keeping distances, not varying is the whole point. One phenomenon, two verdicts — as when Paper 95 separated convention from fact, the verdict is not a property of the phenomenon.
7. The fence: what it concentrates around is not a constant
Observation 7.1 A common misreading is that in high dimension everything settles onto a single value independent of $d$. The maximum of $d$ standard gaussians is a $1$-Lipschitz function and so does concentrate — around a value that keeps rising.
Verification 7.2: $4000$ draws in each dimension.
| $d$ | mean of the max | $\sqrt{2\ln d}$ | ratio | spread over mean |
|---|---|---|---|---|
| $10$ | $1.549540$ | $2.145966$ | $0.7221$ | $0.3830$ |
| $100$ | $2.501465$ | $3.034854$ | $0.8242$ | $0.1721$ |
| $1000$ | $3.238935$ | $3.716922$ | $0.8714$ | $0.1067$ |
| $10000$ | $3.855180$ | $4.291932$ | $0.8982$ | $0.0779$ |
| $100000$ | $4.382860$ | $4.798526$ | $\mathbf{0.9134}$ | $\mathbf{0.0614}$ |
The two columns are not equal — $\sqrt{2\ln d}$ is the leading growth rate rather than the value, and the ratio approaches $1$ slowly. (QED)
Observation 7.3 The spread relative to the mean shrinks from $0.383$ to $0.0614$, so it certainly concentrates. But the value it concentrates on climbs from $1.55$ to $4.38$. “Concentrates” is a statement about the spread, not about the location.
Verification 7.4: And without a Lipschitz bound nothing at all is promised. Take the product of the coordinates, whose Lipschitz constant blows up.
| $d$ | spread over mean | concentrates? |
|---|---|---|
| $2$ | $1.202000$ | no |
| $5$ | $3.119326$ | no |
| $10$ | $10.070772$ | no |
| $20$ | $\mathbf{18.666851}$ | no |
The relative spread grows with $d$. (QED)
Observation 7.5 Concentration is a theorem about Lipschitz functions and not about high dimension as such. The fence is the Lipschitz constant.
8. The census
Observation 8.1
| how it is called | what it is a statement about | quantity | curse or blessing |
|---|---|---|---|
| mass in the skin | inside one body | $1-(1-\varepsilon)^d$ | neither |
| ball loses to cube | two bodies compared | $V_d/2^d$ | grids stop working |
| nearly orthogonal | two random points | $1/\sqrt{d}$ | blessing |
| nearest loses meaning | many random points | relative gap | curse |
| samples grow exponentially | a covering count | $\varepsilon^{-d}$ | curse |
Observation 8.2 These five are not a staircase. The first two are geometric identities and need no randomness. The next two are statements about random points and need probability. The last is a count of an algorithm's work and is not geometry at all. Because one phrase covers them, one remedy seems to be called for.
9. Conclusion
- The unit ball's volume peaks at $d=5$ (Ver. 2.1): $2.37\times10^{-40}$ at $d=100$.
- The solid angle peaks at $d=7$, so even the peaks differ (Obs. 2.2).
- The skin and the cube are different statements (Obs. 3.3).
- Two random directions are nearly orthogonal (Ver. 4.1): $0.0252$ at $d=1000$, matching prediction.
- The nearest neighbour loses meaning (Ver. 5.1): a relative gap of $0.113$.
- The same concentration becomes the blessing (Ver. 6.2): $5.5\%$ mean distortion at tenfold compression.
- What it concentrates on is not a constant (Ver. 7.2): spread down to $0.0614$ while the value climbs to $4.38$.
References
- Lévy, P. (1951). Problèmes concrets d'analyse fonctionnelle. Paris: Gauthier-Villars.
- Bellman, R. (1961). Adaptive Control Processes: A Guided Tour. Princeton: Princeton University Press.
- Johnson, W. B., Lindenstrauss, J. (1984). Extensions of Lipschitz Mappings into a Hilbert Space. Contemporary Mathematics, 26.
- Milman, V. D., Schechtman, G. (1986). Asymptotic Theory of Finite Dimensional Normed Spaces. Berlin: Springer.
- Beyer, K., Goldstein, J., Ramakrishnan, R., Shaft, U. (1999). When Is Nearest Neighbor Meaningful?. Proceedings of the 7th International Conference on Database Theory.
- Ledoux, M. (2001). The Concentration of Measure Phenomenon. Providence: American Mathematical Society.
- Yamagishi, Y. (2026). “Dimension” Is Not One Thing [Paper 94]. Zenodo. 10.5281/zenodo.21758581
- Yamagishi, Y. (2026). Which Objects Know They Are in Four Dimensions? [Paper 114]. Zenodo. 10.5281/zenodo.21963544
- Yamagishi, Y. (2026). What Does the Exponent of $c$ Count? [Paper 116]. Zenodo. 10.5281/zenodo.22003794
- Yamagishi, Y. (2026). Three Distinct Roots of Aggregation [Paper 124]. Zenodo. 10.5281/zenodo.22057728
- Yamagishi, Y. (2026). Measuring Overlap, or Measuring Displacement [Paper 139]. Zenodo. 10.5281/zenodo.22081801
- Yamagishi, Y. (2026). The Order of the Quantifiers Is the Whole Content [Paper 177]. Zenodo. 10.5281/zenodo.22098103
- Yamagishi, Y. (2026). The Multiplication Does Not Decide the Second System [Paper 181]. Zenodo. 10.5281/zenodo.22104155
Reference
Yamagishi, Y. (2026). “The Curse of Dimensionality” Is Not One Curse [Paper 182]. YOI: https://www.yamagishiyuuki.com/yoi/182
DOI: 10.5281/zenodo.22104170
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 182. This paper is a preprint and has not been peer reviewed.