Average measure of intersection of a convex region with its translate

Question

Let $\lambda$ denote the Lebesgue-measure on $\mathbb{R}^n$, and let $C\subset\mathbb{R}^n$ be a convex region.

My question is about $$f(C):=\int_{C} \lambda(C \cap (x + C) ) \mathrm{d} x.$$ How large can $f(C)$ be? Of course, there is the trivial bound $f(C)\leq \lambda(C)^2$ but I would expect more something like $f(C) = O( \lambda(C) )$ at least. This question has probably been answered somewhere, and I suspect that the following might be very well known: for regions $C$ of constant measure, the value of $f(C)$ is maximal if $C$ is an $n$-ball (or something similar). I couldn't find any useful references regarding this question. If someone could help me out with some reference, or even a simple solution, I would be very grateful.

Answer 1

The following proposition answers OP's question regarding the upper bound of $$\tau(C) \Doteq f(C)/\lambda^2(C).$$

Let $B_n$ be the closed Euclidean unit ball of $\mathbb{R^n}$ centred at $0$, that is $$B_n = \{ (x_1,\dots, x_n) \in \mathbb{R^n} \, \vert \,\, x_1^2 + \cdots + x_n^2 \le 1\},$$ and let $\tau_n = \tau(B_n)$.

Proposition. The Euclidean unit ball $B_n \subset \mathbb{R}^n$ maximizes $\tau(C)$ among all convex subsets of finite and positive Lebesgue measure, i.e., $\tau(C) \le \tau_n$ holds for any such $C$.

Claim 3 below gives a way to compute efficiently $\tau_n$ for all $n > 0$.

The proof of the proposition follows Fedja's idea. The first part consists in proving that $\tau$ doesn't decrease after [Steiner symmetrization][2]. This is Claim 4 below. The second part uses [1, Theorem 7.1.10] showing that some Euclidean ball centred at $0$ can be obtained as the limit of Steiner symmetrizations of $C$ for any compact convex $C$ of positive Lebesgue measure (equivalently, with non-empty interior).

When $n = 1$, the result is slightly stronger.

Claim 1. Let $C \subset \mathbb{R}$ be a convex subset of finite Lebesgue measure $\lambda(C) > 0$. Then $\tau(C) \le \frac{3}{4}$ and equality holds if and only if $C$ is centred at $0$.

So, the above claim settles OP's question when $n = 1$. Unsurprisingly, the result generalizes to $\mathbb{R}^n$ in the following way.

Claim 2. Let $C \subset \mathbb{R}^n$ be a parallelotope of positive Lebesgue measure. Then $\tau(C) \le (\frac{3}{4})^n$ and equality holds if and only if $C$ is centred at $0$.

Computing the maximum $\tau_n$ of $\tau$ can be achieved via the formula

Claim 3. We have $$ \tau_n = \frac{3}{2} \frac{\int^{\pi/3}_0 \sin^n(\theta) d\theta}{\int^{\pi/2}_0 \sin^n(\theta) d\theta}.$$ In particular, $\tau_1 = \frac{3}{4}, \, \tau_2 = 1 - \frac{3 \sqrt{3}}{4 \pi}, \tau_3 = \frac{15}{32}, \, \tau_4 = 1 - \frac{9 \sqrt{3}}{8 \pi}, \tau_5 = \frac{159}{512}, \, \tau_6 = 1 - \frac{27 \sqrt{3}}{20 \pi} $.

The above formula can be rephrased as $\tau_n = \frac{3}{2}\frac{W_n'}{W_n}$ where $W_n$ is the $n$-th [Wallis' integral][3] and $W_n'$ is recursively defined by $W'_0 = \frac{\pi}{3}, \,W_1' = \frac{1}{2}$ and $W_n' = - \frac{1}{2n}(\frac{\sqrt{3}}{2})^{n - 1} + \frac{n - 1}{n}W'_{n - 2}$. The latter formula allows an efficient computation of $\tau_n$ for any $n > 0$. As $W_n' \le \frac{\pi}{3}(\frac{\sqrt{3}}{2})^n$ and $W_n \sim \sqrt{\frac{\pi}{2n}}$, the sequence $(\tau_n)_n$ converges to $0$ exponentially fast (we have for instance $10^{-7} < \tau_{91} < 10^{-6}$).

Before proving the claims, let us state some obvious, but important, remarks.

Remark 1. Let $A \in GL_n(\mathbb{R})$ and $d \in \mathbb{R}^n$. Let $C$ be any subset of $\mathbb{R}^n$. Then we have $$f(AC) = \vert \det(A) \vert^2f(C), \quad f(C + d) = \int_C \lambda\left(C \cap (C + d + x) \right)d\lambda(x).$$

In particular the ratio $\tau(C)$ is $GL_n(\mathbb{R})$-invariant but it is not invariant under translation. Indeed, if $C$ is a convex subset of finite positive Lebesgue measure whose interior contains $0$, then the function $f_C(d): d \mapsto f(C + d)$ has a non-empty and bounded support.

The following tightly relates to OP's question.

Question. Let $C \subset \mathbb{R}^n $ be a convex subset of finite positive Lebesgue measure. For which $d$ is $f_C(d)$ maximal?

The following relates $\tau$ to Steiner symmetrization.

Claim 4. Let $C \subset \mathbb{R}^n $ be a convex subset of finite positive Lebesgue measure. Let $H =\{0\} \times \mathbb{R}^{n -1}$ and let $C' \Doteq \text{St}_H(C)$ be the [Steiner symmetrization][2] of $C$ with respect to $H$. Then we have $\tau(C) \le \tau(C')$.

We now turn to the proofs of the above claims.

Proof of Claim 1. Because of Remark 1, we can restrict, without loss of generality, to intervals of the form $[-1, 1] + d$ with $d \ge 0$. A direct computation yields $$ f([-1, 1] + d) = \int_{-1}^1 (2 - \vert d + x \vert)^{+} dx = \left\{\begin{array}{cc} 3 -d^2 & \text{ if } 0 \le d \le 1, \\ \frac{1}{2}(3 - d)^2 & \text{ if } 1 \le d \le 3, \\ 0 & \text{ if } d \ge 3. \end{array}\right. $$ where $g^{+} \Doteq \max(g ,0)$ for $g$ a real-valued function. The result follows.

Proof of Claim 2. By Remark 1, we can restrict to the convex sets of the form $C = [-1, 1]^n + d$ with $d = (d_1, \dots, d_n) \in \mathbb{R}^n$. We have $f(C) = \int_{[-1, 1]^n} (2 - \vert x_1 + d_1 \vert)^{+} \cdots (2 - \vert x_n + d_n\vert)^{+}dx_1 \cdots dx_n$. It follows from Claim 1, that $f(C) \le \int_{[-1, 1]^n} (2 - \vert x_1 \vert)^{+} \cdots (2 - \vert x_n \vert)^{+}dx_1 \cdots dx_n = 3^n$ and the inequality is strict whenever one of the $d_i$'s is not zero. The result follows.

We denote by $V_n(r)$ the Lebesgue measure of the Euclidean ball of radius $r \ge 0$, that is, $V_n(r) = r^n\frac{\pi^{\frac{n}{2}}}{\Gamma(\frac{n}{2} + 1)}$, and let $\nu_n = V_n(1) = \lambda(B_n)$. Given a real number $h$ such that $-1 \le h \le 1$, we denote by $B_n^{\text{cap}}(h)$ the hyperspherical cap of height $h$ along $x_n$ , that is the set of points $x = (x_1, \dots, x_n) \in B_n$ such that $x_n \ge h$. We also set $V_n^{\text{cap}}(h) = \lambda(B_n^{\text{cap}}(h))$.

Proof of Claim 3. By definition, we have $f(B_n) = \int_{B_n} \lambda(B_n \cap (B_n + x)) d\lambda(x) = 2 \int_{B_n} V_n^{\text{cap}}(\frac{\vert x \vert}{2}) d\lambda(x)$ where $\vert \cdot \vert$ stands for the Euclidean norm of $\mathbb{R}^n$. Changing to [$n$-dimensional spherical coordinates][4], we get $$f(B_n) = 2 \int_{D_n} V_n^{\text{cap}}(\frac{r}{2})r^{n - 1} \sin^{n - 2}(\phi_1)\sin^{n - 3}(\phi_2) \cdots \sin(\phi_{n - 1}) dr d\phi_1 d\phi_2 \cdots d \phi_{n - 1}$$ where $D_n = [0,2\pi] \times [0, \pi] \times \cdots \times [0, \pi] \times [0, 1]$. Using the classical computation of $\nu_n = \lambda(B_n)$, we deduce that $$f(B_n) = 2n\nu_n \int_{[0, 1]} V_n^{\text{cap}}(\frac{r}{2})r^{n - 1}dr = 2^{n + 1}n\nu_n \int_{[0, \frac{1}{2}]} V_n^{\text{cap}}(s)s^{n - 1}ds .$$ Since $V_n^{\text{cap}}(s) = \int_s^1 V_{n - 1}(\sqrt{1 - h^2})dh = \nu_{n - 1} \int_0^{\text{arcos}(s)} \sin^n(\theta)d \theta$, we obtain $$f(B_n) = 2^{n + 1}n\nu_n \nu_{n - 1} \int_{[0, \frac{1}{2}]} J_n(\text{arcos}(s))s^{n - 1}ds$$ where $J_n(\phi) = \int_0^{\phi}\sin^n(\theta)d\theta$. Using the substitution $\theta = \text{arcos}(s)$, we get $$ \begin{eqnarray} \frac{f(B_n)}{2^{n + 1}n\nu_n \nu_{n - 1}} &=& \int_{[\frac{\pi}{3}, \frac{\pi}{2}]} J_n(\theta)\cos(\theta)^{n - 1} \sin(\theta)d\theta \\ &=& \frac{[J_n(\theta) \cos^n(\theta)]^{\frac{\pi}{3}}_{\frac{\pi}{2}} + \int^{\frac{\pi}{2}}_{\frac{\pi}{3}} (\cos(\theta)\sin(\theta))^n d\theta}{n} \\ &=& \frac{2J_n(\frac{\pi}{3}) + \int^{\pi}_{\frac{2\pi}{3}} \sin(\theta)^n d\theta}{2^{n + 1} n} \\ &=& \frac{3J_n(\frac{\pi}{3})}{2^{n + 1} n}. \end{eqnarray} $$ Now the result follows from the identity $\frac{\nu_{n - 1}}{\nu_n} = \frac{1}{2 W_n}$.

Claim 4 is a specialization of

Fedja's Lemma. Let $$\mathcal{I}(C, D, E) \Doteq \int_C \lambda(D \cap (E + x)) d\lambda(x)$$ where $C$, $D$ and $E$ are convex subsets of $\mathbb{R}^n$ of positive finite Lebesgue measure. Then $\mathcal{I}(C, D, E) \le \mathcal{I}(C', D', E')$, where $X'$ denotes the Steiner symmetrization of $X$ with respect to $H = \{0\} \times \mathbb{R}^{n - 1}$.

The proof of Fedja's Lemma can be reduced to the case $n = 1$. The proof of the case $n = 1$ is eased off by

Lemma 1. Let $g: \mathbb{R} \rightarrow \mathbb{R}^{+} \cup \{0\}$ be an even non-negative function which is concave on its support and non-decreasing on the negative real numbers. Then we have $$\int_I g d\lambda \le \int_{I'} g d\lambda $$ for every convex subset $I$ of $\mathbb{R}$ where $I'$ is the closed interval of measure $\lambda(I)$ centred at $0$.

Proof of Lemma 1. The non-negative number $\int_I g d\lambda$ is the area of convex subset $C(I) \subset \mathbb{R}^2$ located between the $x$-axis and the graph of $g$. By hypothesis, we have $C(I)' \subset C(I') = \int_{I'} g d\lambda$, where $C(I)'$ is Steiner symmetrization of $C(I)$ with respect to $H = \{0\} \times \mathbb{R}$. As $C(I)$ and $C(I)'$ have the same area, the result follows.

We are now in position to prove Fedja's Lemma.

Proof of Fedja's Lemma. Let us assume first that $n = 1$. As $\mathcal{I}(C + x, D, E) = \mathcal{I}(C, D, E + x)$ and $\mathcal{I}(C, D + x, E + x) = \mathcal{I}(C, D, E)$, we can assume, without loss of generality, that $C$ and $D$ are centred at zero. Let us write $E = E’ + \eta$ and let $g(x) \Doteq \lambda(D \cap (E' + x))$. Then we have $$ g(x) = \left\lbrace \begin{array}{ccc} \min(d, e) & \text{ if } \vert x \vert \le \frac{ \vert d - e \vert }{2}, \\ \frac{d + e}{2} - \vert x \vert & \text{ if } \frac{ \vert d - e \vert }{2} \le \vert x \vert \le \frac{d + e}{2}, \\ 0 & \text{ if } \vert x \vert \ge \frac{d + e}{2}, \end{array}\right. $$ where $d = \lambda(D) > 0$ and $e = \lambda(E) > 0$. The function $g$ satisfies the hypothesis of Lemma $1$, thus $$ \mathcal{I}(C, D, E) = \mathcal{I}(C + \eta, D, E’) = \int_{C + \eta} g(x) d\lambda(x) \le \int_{C} g(x) d\lambda(x) = \mathcal{I}(C, D, E’)$$ which proves the result for $n = 1$. Assume now that $n > 1$. We denote by $\lambda_{n - 1}$ the Lebesgue measure on $\mathbb{R}^{n - 1}$ and we identify $H$ with $\mathbb{R}^{n - 1}$. For $\xi \in \mathbb{R^n}$, we write $\xi = (\xi_1, \zeta)$, with $\xi_1 \in \mathbb{R}$ and $\zeta \in H$. Given $A \subset \mathbb{R}^n$ and $\zeta \in H$, we set $A^{\zeta} \Doteq \{ \xi_1 \, \vert \, (\xi_1, \zeta) \in A\} \subset \mathbb{R}$. Let $x = (x_1, z) \in \mathbb{R} \times H$. Observe that $(A^{\zeta})’ = (A’)^{\zeta}$ and $(A + x)^{\zeta} = A^{\zeta - z} + x_1$. By the Fubini-Tonelli Theorem, we have $\lambda_n(D \cap (E + x)) = \int_H \lambda_1(D^{\zeta} \cap (E^{\zeta - z} + x_1)) d \lambda_{n - 1}(\zeta)$. We shall use the abbreviations $dz = d \lambda_{n - 1}(z)$ and $d\zeta = d \lambda_{n - 1}(\zeta)$. Using Fubini-Tonelli’s Theorem and the case $n = 1$, we obtain $$ \begin{eqnarray} \mathcal{I}(C, D, E) & = & \int_{H^2} \mathcal{I}(C^{z}, D^{\zeta}, E^{\zeta - z})d\zeta dz \\ & \le & \int_{H^2} \mathcal{I}((C’)^{z}, (D’)^{\zeta}, (E’)^{\zeta - z}) d\zeta dz \\ & = & \mathcal{I}(C’, D’, E’). \end{eqnarray} $$

Eventually, we can prove the proposition.

Proposition's proof. By [1, Theorem 7.1.10], there is a Euclidean ball $B$ centred at zero which is the limit in the Hausdorff distance topology of Steiner symmetrizations of $C$ with respect to linear hyperplanes. As $\lambda$, $f$ and hence $\tau$ are continuous with respect to the Hausdorff distance topology (we use that $C$ has non-empty interior), we have $\lambda(B) = \lambda(C) > 0$ and we infer from Claim 4 that $\tau(C) \le \tau(B)$. By Remark 1, we have $\tau(B) = \tau_n$, which completes the proof.

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[1] S. G. Krantz, H. R. Parks, "The Geometry of Domains in Space", 1999.
[2]: https://en.wikipedia.org/wiki/Symmetrization_methods [3]: https://en.wikipedia.org/wiki/Wallis%27_integrals [4]: https://en.wikipedia.org/wiki/N-sphere

Answer 2

How about a short solution to a much general problem, and without convexity assumption (which turns out to be really just a distraction here) ?

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So, more generally, for nonempty compact subsets $A$, $B$, and $C$ of $\mathbb R^n$, define $F(A,B,C)$ by

$$ F(A,B,C) := \int_{A}\lambda(B \cap (x + C))dx. $$

In particular, the OP's question is concerned with $f(C) := F(C,C,C)$.

We shall proceed in 3 short steps.

• Step 1. The first step in analyzing $F$ is to realize that we can rewrite it as $F(A,B,C) = G(1_A,1_B,1_C)$, where

\begin{align} G(u,v,w) = \int_{\mathbb R^n}\int_{\mathbb R^n} u(x)w(x-y)v(y)\,dxdy. \end{align}

• Step 2. Now, by the Riez rearrangement inequality, we know that $$ G(u,v,w) \le G(u^\star,v^\star,w^\star), $$ where $u^\star$ is the Symmetric decreasing rearrangement of the function $u$. On the other hand, it is clear that $$ (1_A)^{\star} = 1_{K(A)}, $$ where $K(A)$ is the centered ball of same volume as $A$. We deduce that $$ F(A,B,C) \le F(K(A),K(B),K(C)), $$

and in particular, $f(C) \le f(K(C))$. The solution to the OP's question then follows from computation for the ball, in the begining of the accepted answer https://mathoverflow.net/a/282941/78539.

Answer 3

Analytic formula and an exponential bound

For $r \in [0, 1]$, define $\tau_m(r) \in [0, 1]$ by $$ \tau_m(r) := \frac{1}{\lambda(B_m(r))}\int_{B_m(r)} \frac{\lambda(B_m(1) \cap (B_m(1) + x))}{\lambda(B_m(1))}dx. $$

The case $r=1$ has been analysed in accepted answer. Extending user @Matt's comment, one can establish that (for the core idea, see this post)

$$ \begin{split} \tau_m(r) &= \frac{1}{V_m^{cap}(r, 0)}\int_{0}^r ms^{m-1}V_m^{cap}(1, s/2)ds = \frac{1}{r^mV_m^{cap}(1, 0)}\int_{0}^r ms^{m-1}V_m^{cap}(1, s/2)ds\\ &= \frac{1}{V_m^{cap}(1, 0)}\int_{0}^1 ms^{m-1}V_m^{cap}(1, rs/2)ds, \end{split} $$

where $V_m(r; h)$ is the volume (i.e Lebesgue measure) of the half-lens $\{x \in B_m(r) \mid x_1 \ge h\}$.

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Preamble. I propose to obtain an analytic formula for $\tau_m(r)$ in terms of special functions (mostly hypergeometric functions). We will also obtain a good upper bound for $\tau_m(r)$.

The end result will be the following exponential decay of $\tau_m(r)$ as a function on the dimension $m$.

Theorem. For every $r \in [0, 1]$ and large integer $m$, we have the upper bound $$ \begin{split} \tau_m(r) &\lesssim \exp\left(-\frac{mr^2}{8-r^2}\right) + \frac{r(1-r^2/4)^{m/2}}{\sqrt{m\pi}}\\ &\lesssim \exp\left(-\frac{mr^2}{8}\right) + \frac{r(1-r^2/4)^{m/2}}{\sqrt{m\pi}}. \end{split} $$

Proof. Now, it is a classical computation that for $h \ge 0$ we have $V_m^{cap}(1, h) = \frac{\lambda(B_{m - 1}(1))}{2}\beta_{1-h^2}\left(\frac{m+1}{2},\frac{1}{2}\right)$, where $\beta_x(a, b) := \int_0^x t^{a-1}(1-t)^{b-1}dt$ defines the incomplete beta function. In particular, $V_m^{cap}(1, 0) = \frac{\lambda(B_{m - 1}(1))}{2}\beta_{1}\left(\frac{m+1}{2},\frac{1}{2}\right)$.

Thus, integrating by parts, we get

$$ \begin{split} V_m^{cap}(1,0)\tau_m(r) &= \int_{0}^1 ms^{m-1}V_m^{cap}(rs/2)ds = \int_{0}^1 \beta_{1-(rs)^2/4}\left(\frac{m+1}{2},\frac{1}{2}\right)ds^m\\ &= \left[s^m\beta_{1-(rs)^2/4}\left(\frac{m+1}{2},\frac{1}{2}\right)\right]_0^1 + r\int_0^1 s^m(1-(rs)^2/4)^{\frac{m-1}{2}}ds \\ &= \beta_{1-(r/2)^2}\left(\frac{m+1}{2},\frac{1}{2}\right) + R_m, \end{split} \tag{1} $$ where $R_m := r\int_0^1 s^m(1-(rs/2)^2)^{\frac{m-1}{2}}ds$. Now, consider the change of variable $s = 2\sqrt{t}$. This gives $s^m(1-s^2/4)^{(m-1)/2}ds=2^mt^{m/2}\cdot(1-t)^{(m-1)/2}\cdot t^{-1/2}dt = 2^mt^{(m-1)/2}(1-t)^{(m-1)/2}dt$. Thus,

$$ \begin{split} R_m &= r\int_0^1 s^m(1-(rs)^2/4)^{\frac{m-1}{2}}ds = \frac{2^m}{r^m}\int_0^{(r/2)^2}t^{(m-1)/2}(1-t)^{(m-1)/2}dt\\ &=: \frac{2^m}{r^m}\beta_{(r/2)^2}\left(\frac{m+1}{2},\frac{m+1}{2}\right). \end{split} \tag{2} $$

Combining (1) and (2) then gives the analytic formula,

Analytic formula. For any $r \in [0, 1]$ and positive integer $m$, it holds that $$ \tau_m(r) = I_{1-(r/2)^2}\left(\frac{m+1}{2},\frac{1}{2}\right) + \frac{(2/r)^m\beta_{(r/2)^2}\left(\frac{m+1}{2},\frac{m+1}{2}\right)}{\beta_1\left(\frac{m+1}{2},\frac{1}{2}\right)}, $$ where $u \mapsto I_u(a, b) := \beta_u(a,b) / \beta_1(a, b)$ is the CDF of $(a,b)$-beta distribution.

Bounding the first term. Let $\delta := (r/2)^2$ and $p = I_{1-\delta}((m+1)/2,1/2)$. It is known since Temme (1992) that for $p \in (0, 1)$ and large $a > 0$, the solution of the equation $p = I(t; a,b)$ is given (approximately) by

$$ t=t_p(a, b) \approx e^{-(1/a)Q_{1-p}(\Gamma(b,1))}, \tag{4} $$

where $Q_{1-p}(\Gamma(b,1))$ is the $1-p$ quantile of the unit-scale gamma distribution with shape parameter $b$. Now by standard concentration results (e.g see Boucheron et al. textbook),

$$ Q_{1-p}(\Gamma(b,1)) \le \log(1/p) + \sqrt{2b\log(1/p)}. \tag{5} $$

In particular, for $a=(m+1)/2$ and $b=1/2$ we get

$$ Q_{1-p}(\Gamma(1/2,1)) \le \log(1/p) + \sqrt{\log(1/p)}. \tag{6} $$

Putting (2), (4), and (6) together and using the basic inequality $1-e^{-z} \le 2z/(2+z)\;\forall z \ge 0$, we see that

$$ \begin{split} \delta &= 1 - t_{p}\left((m+1)/2,1/2\right) \approx 1-e^{-\frac{2Q_{1-p}(\Gamma(1/2,1))}{m+1}} \le 1-e^{-\frac{2}{m+1}\left(\log\left(\frac{1}{p}\right) + \sqrt{\log\left(\frac{1}{p}\right)}\right)} \le \frac{2\alpha}{2+\alpha}, \end{split} $$

where $\alpha = \frac{2}{m+1}(\log(1/p)+\sqrt{\log(1/p)})$. Thus, $\alpha \ge \delta/(1 - \delta/2)$, and this wolfram-alpha query tells us that we must have $p \lesssim \exp(-mr^2/(8-r^2))$. Putting things together then gives

$$ \begin{split} I_{1-(r/2)^2}((m+1)/2,1/2) = p \lesssim \exp(-mr^2/(8-r^2)) \le \exp(-mr^2/8). \end{split} $$

Bounding the second term. It is a well-known result that

$$ \beta_z(a,b) \equiv a^{-1}z^a{}_2F_1(a,1-b,a+1;z), $$

where $z \mapsto {}_2F_1(u, v, w; z)$ is the hypergeometric function. Thus, one computes

$$ \begin{split} (2/r)^m\beta_{(r/2)^2}\left(\frac{m+1}{2},\frac{m+1}{2}\right) &= \frac{r}{m+1}{}_2F_1\left(\frac{m+1}{2},\frac{1-m}{2},\frac{m+3}{2},\frac{r^2}{4}\right) \\ &= \frac{r}{m+1}{}_2F_1\left(\frac{1-m}{2},\frac{m+1}{2},\frac{m+3}{2},\frac{r^2}{4}\right)\\ &\approx \frac{r}{m+1}{}_2F_1\left(\frac{1-m}{2},\frac{m+1}{2},\frac{m+1}{2},\frac{r^2}{4}\right)\\ &= \frac{r}{m+1}(1-r^2/4)^{(m-1)/2}, \end{split} $$

where we have used the powerful identity ${}_2F_1(u, v, v; z) \equiv (1 - z)^{-u}$. On the other hand, Stirling's formula gives

$$ \beta_1\left(\frac{m+1}{2},\frac{1}{2}\right) \approx \Gamma(1/2)\left(\frac{m+1}{2}\right)^{-1/2} = \sqrt{\frac{\pi}{m+1}}. $$

Thus,

$$ \frac{(2/r)^m\beta_{(r/2)^2}\left(\frac{m+1}{2},\frac{m+1}{2}\right)}{\beta_1\left(\frac{m+1}{2},\frac{1}{2}\right)} \approx \frac{r(1-(r/2)^2)^{(m-1)/2}}{\sqrt{\pi(m+1)}} $$

Putting every together then completes the proof of the theorem. $\quad\quad\quad\Box$