TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passaudited 2026-08-24

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Closed Euclidean balls are Jordan measurable and their volumes satisfy the slicing recursion

Statement

For every integer $n \geq 1$ and every $r \geq 0$ , put ${\overline{B}}_{n} (0, r) : = {x \in R^{n} : ∥ x ∥_{2} \leq r}$ , extending the positive-radius notation of Euclidean spheres and closed balls as subspaces of $R^{n}$ to $r = 0$ . This closed Euclidean ball is Jordan measurable; write its content as $V_{n} (r)$ . One has $V_{1} (r) = 2 r$ . For $n \geq 2$ and $r \geq 0$ , $V_{n} (r) = V_{n - 1} (1) \int_{- r}^{r} (r^{2} - t^{2})^{(n - 1) / 2} d t$ .

Facts & Assumptions

Given: Positive integer dimension $n$ , radius $r \geq 0$ , and the closed Euclidean balls defined in the Statement.

[F1]

A solid between continuous graphs over a compact Jordan base is compact and Jordan measurable (A solid between continuous graphs over a compact Jordan base is Jordan measurable and integrates by vertical sections).

[F2]

If a linear map $T$ has matrix $A$ and $E$ is a bounded Jordan set, then $cont (T (E)) = ∣ \det A ∣ cont (E)$ (A linear endomorphism of $R^{n}$ sends bounded Jordan sets to bounded Jordan sets and scales their content by the absolute determinant).

[F3]

If a bounded Jordan set has Jordan-measurable sections outside a content-zero parameter set, then its completed sectional-content function, with empty sections assigned content $0$ , is integrable and its integral is the set's content (Cavalieri: Jordan content is the integral of sectional contents, and equal sections give equal content).

[F4]

A composite of continuous maps is continuous (Continuity may be checked on any open cover, and on any finite closed cover; composites of continuous maps are continuous).

[F5]

The inverse of a continuous injective real function on an interval is continuous on its image (Continuous inverse theorem: a continuous injective $f$ on an interval $I$ is a bijection onto the order-convex set $f [I]$ , and the inverse $g : f [I] \to I$ is continuous and strictly monotone in the same sense as $f$ ).

[F6]

Every nonnegative real has a unique nonnegative square root (Existence and uniqueness of $n$ -th roots: a unique $a^{1 / n} \geq 0$ with $(a^{1 / n})^{n} = a$ ).

[F7]

A Euclidean set is compact if and only if it is closed and bounded (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line).

[F8]

The Euclidean distance is $d_{2} (u, v) = ∥ u - v ∥_{2}$ and satisfies the metric triangle inequality ( $R^{n}$ as the set of functions $n \to R$ , and $d_{1}$ , $d_{2}$ , $d_{\infty}$ are metrics on it).

Proof

technique · induction

1.1givenF7base

For $n = 1$ , the ball is the closed bounded interval $[- r, r]$ , hence compact by [F7], and it is Jordan measurable with content $2 r$ , including $r = 0$ .

2.1step 1.1ihF1F4F5F6F8algebra

Assume closed balls in dimension $n - 1$ are compact and Jordan measurable. The triangle inequality in [F8], applied in both orders, gives $∣ ∥ u ∥_{2} - ∥ v ∥_{2} ∣ \leq ∥ u - v ∥_{2}$ , so the Euclidean norm is continuous. On $[0, r]$ , the estimate $∣ (r^{2} - s^{2}) - (r^{2} - t^{2}) ∣ = ∣ s - t ∣ ∣ s + t ∣ \leq 2 r ∣ s - t ∣$ makes $s \mapsto r^{2} - s^{2}$ continuous; [F5] and [F6] make its nonnegative square root continuous, and [F4] makes the resulting composite continuous on the ball. Thus the $n$ -ball is the solid between two continuous square-root graphs over ${\overline{B}}_{n - 1} (0, r)$ , and [F1] makes it compact and Jordan measurable.

3.1step 2.1F2algebra

For $∣ t ∣ \leq r$ , the section at last coordinate $t$ is the $(n - 1)$ -ball of radius $ρ (t) : = \sqrt{r^{2} - t^{2}}$ . It is the image of the unit $(n - 1)$ -ball under scalar multiplication by $ρ (t)$ , whose determinant has absolute value $ρ (t)^{n - 1}$ ; [F2] gives section content $V_{n - 1} (1) (r^{2} - t^{2})^{(n - 1) / 2}$ . At $t = \pm r$ this is zero.

4.1step 3.1F3discharge-induction∎

By [F3], integration of the section contents in step 3.1 gives the displayed recursion. Thus the induction proves Jordan measurability in every positive dimension and the recursion for every $n \geq 2$ .

Depends on

Used by

The closed form for the volume of the unit n-ball Corollary
The volume of a radius-r closed n-ball is π^n/2rⁿ/Γ(n/2+1) Corollary
A compact-support Cauchy–Pompeiu calculation Example
Slicing gives the unit-ball volumes through dimension five Example
The Bochner–Martinelli formula for C1 functions Theorem

Dependency tree · two levels

95 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

University of Toronto MAT237Y1, The Gamma Function and the Beta Function, §2.4 (standard reference, not scraped)
Sheldon Axler, Measure, Integration & Real Analysis, §5C (standard reference, not scraped)