论文标题
最大化投影空间中点对点之间角度的预期能力
Maximizing expected powers of the angle between pairs of points in projective space
论文作者
论文摘要
Among probability measures on $d$-dimensional real projective space, one which maximizes the expected angle $\arccos(\frac{x}{|x|}\cdot \frac{y}{|y|})$ between independently drawn projective points $x$ and $y$ was conjectured to equidistribute its mass over the standard Euclidean basis $ \ {e_0，e_1，\ ldots，e_d \} $ byfejesTóth\ cite {ft59}。如果是真的，那么这个猜想显然意味着相同的措施最大化$ \ arccos^α的期望（\ frac {x} {| x |} \ cdot \ cdot \ frac {y} {y | y |}）$对于任何指数$α> 1 $。内核$ \ arccos^α（\ frac {x} {| x |} \ cdot \ frac {y} {y | y |}）$代表了无限维二次程序的目的。我们在非空范围内验证了此{温和}的离散版本$α>α__{δ^d} \ ge 1 $，并确定所得最大化器$ \ hatμ$的独特性。当$α<α_{δ^d} $时，我们显示$ \hatμ$不再最大化。在此范围的端点$α=α_{δ^d} $上，我们还必须显示另一个最大化器$μ$，这不是$ \ hatμ$的旋转。对于猜想的连续版本，Bilyk等人对此工作的早期草稿提供了附录，结合了当前的改进，以产生$α_{δ^d} <2 $。原始猜想$ \ ald = 1 $保持开放（除非$ d = 1 $）。 但是，在最大可能的范围$α> 1 $的情况下，我们显示$ \hatμ$，其旋转量最大化上述期望在$ l^\ infty $ -Kantorovich-kantorovich-rubinstein-Wasseint the Off From from Optial Fromplation the $ l^\ infty $ -Kantorovich $ -Kantorovich-rubinstein-wastein $ d_ \ inftty;对于任何度量$μ$，相对于$ \ hatμ$都是相同的，但球的大小取决于{$α，d $和} $ \ | \ | \ | \ frac {d \ hatμ} {dμ} {dμ} {dμ} {dμ} \ | _ | _ {\ iffty} $。
Among probability measures on $d$-dimensional real projective space, one which maximizes the expected angle $\arccos(\frac{x}{|x|}\cdot \frac{y}{|y|})$ between independently drawn projective points $x$ and $y$ was conjectured to equidistribute its mass over the standard Euclidean basis $\{e_0,e_1,\ldots, e_d\}$ by Fejes Tóth \cite{FT59}. If true, this conjecture evidently implies the same measure maximizes the expectation of $\arccos^α(\frac{x}{|x|}\cdot \frac{y}{|y|})$ for any exponent $α> 1$. The kernel $\arccos^α(\frac{x}{|x|}\cdot \frac{y}{|y|})$ represents the objective of an infinite-dimensional quadratic program. We verify discrete and continuous versions of this {milder} conjecture in a non-empty range $α> α_{Δ^d} \ge 1$, and establish uniqueness of the resulting maximizer $\hat μ$ up to rotation. We show $\hat μ$ no longer maximizes when $α<α_{Δ^d}$. At the endpoint $α=α_{Δ^d}$ of this range, we show another maximizer $μ$ must also exist which is not a rotation of $\hat μ$. For the continuous version of the conjecture, an appendix provided by Bilyk et al in response to an earlier draft of this work combines with the present improvements to yield $α_{Δ^d}<2$. The original conjecture $\ald=1$ remains open (unless $d=1$). However, in the maximum possible range $α>1$, we show $\hat μ$ and its rotations maximize the aforementioned expectation uniquely on a sufficiently small ball in the $L^\infty$-Kantorovich-Rubinstein-Wasserstein metric $d_\infty$ from optimal transportation; the same is true for any measure $μ$ which is mutually absolutely continuous with respect to $\hat μ$, but the size of the ball depends on {$α,d$, and} $\|\frac{d \hat μ}{dμ}\|_{\infty}$.
