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Let $X$ be a random variable on $\mathbb R^n$ and let $S_p^n := \{w \in \mathbb R^n \mid \|w\|_p = 1\}$ be the unit-sphere w.r.t to the $\ell_p$-norm in $\mathbb R^n$. We will be particularly interested in the case $p \in \{1,2\}$. For any $w \in B_p^n$ and $t \ge 0$, let $$ L(X'w,t):= \sup_{u \in \mathbb R}\mathbb P(|X' w - u| \le t). $$ be the Lévy concentration function of the random variable $X' w$. Finally, define $$ L(S_p^n,t) := \sup_{w \in S_p^n} L(X'w,t). $$

Question. Under what minimal assumptions on $X$ is it true that $\lim_{t \to 0^+} L(S_p^n, t) \to 0 $ ?

For example, does it suffice to assume that

• $X$ has density which is sufficiently smooth (e.g continuous) ?
• What if we simply assume that the distribution of $X$ is atomless ?

Some solved cases

• (1) Bounded Radon transform of density of $X$. Suppose there exists a positive constant $b$ such that for every $w \in B_p^n$, the random variable $X'w$ has density bounded by $b$. Then, $$ L(S_p^n,t) = \sup_{w \in S_p^n}\sup_{u \in \mathbb R}\mathbb P(|X'w-u| \le t) \le b\cdot 2t = 2bt \to 0. $$ For example, this is the case when the distribution of $X$ is a mixture of multivariate Gaussians $\sum_k\pi_kN(\mu_k,\Sigma_k)$; in this case each $X'w$ has distribution $\sum_k \pi_k N(\mu_k'w,w'\Sigma_k w)$, and the condition clearly holds with $b=1$. Finally, note that if $f$ is the density of $X$, then the density $\rho_w$ of $X'w$ evaluated at a point $c$ is nothing but the Radon transform $R[f]$ of $f$ w.r.t the hyperplane $H_{w,c} := \{x \in \mathbb R^n \mid x'w = c\}$. Thus, the assumption that $\sup_{w \in S_p^n}\|\rho_w\|_\infty \le b$, is really reminiscent of demanding that $R[f] \in L^\infty$. So the question now is, under what minimal conditions if the Radon transform of a density function bounded ?

• (2) $X$ admits a vanishing Small-ball probability. Let us now show that $L(S_p^n,t) \to 0$ if $X$ admits small-ball probability which vanishes at zero, i.e $$ \tag{1} L(X,t) := \sup_{x \in \mathbb R^n}\mathbb P(\|X-x\|_2 \le t) \to 0\text{ in the limit }t \to 0^+. $$ In particular, this is the case when $X$ has bounded density $f$, since $$ L(X,t) \le \|f\|_\infty\cdot \sup_{x\in \mathbb R^n}\mbox{vol}(tB_2^n + x) \le C_n\cdot t^n \to 0. $$ Now, define $\kappa_{n,p} := \inf\{\|w\|_2 \mid w \in B_p^n\} > 0$. It is easy to see that $\kappa_{n,p} = 1/n^{(1/p-1/2)_+}$. For example, $\kappa_{n,1} = 1/\sqrt{n}$ and $\kappa_{n,2} = 1$. Under the condition (1), one then computes $$ \begin{split} L(S_p^n,t) &= \sup_{w \in S_p^n}\sup_{u \in \mathbb R}\mathbb P(|X'w-u| \le t) \le \sup_{w \in S_p^n}\sup_{x \in \mathbb R^n}\mathbb P(|X'w-x'w| \le t)\\ & \le \sup_{w \in S_p^n}\sup_{x \in \mathbb R^n}\mathbb P(\|w\|_2\|X-x\|_2 \le t)\\ &\le \sup_{x \in \mathbb R^n}\mathbb P(\|X-x\|_2 \le t/\kappa_{n,p}),\text{ since }\|w\|_2 \ge \kappa_{n,p}\text{ if }\|w\|_p = 1\\ &= L(X,t/\kappa_{n,p}) \to 0,\text{ by (1)} \end{split} $$

• (3) $X$ is an transformation of log-concave random vector. C

onsider the scenario where $X \overset{D}{=}AZ + \mu$, where $A$ is a deterministic $m \times n$ matrix, $\mu$ is a determistic vector in $\mathbb R^m$, and $Z$ is random variable which is isotropic, log-concave, whose coordinates are $b$-subGaussian, then thanks to Grigoris Paouris' Small Ball Probability Estimates for Log-Concave Measures, we know that $$ L(X'w,t) = L((Aw)'X,t) \le (t/\|Aw\|_2)^{(c/b)^2},\text{ for sufficiently small }t $$ where $c$ is an absolute positive constant. Thus, if $A$ is non-degenerate in the sense that $\kappa_p(A) := \inf_{w \in S_p^n} \|Aw\|_2 \ge 1/C > 0$, then

$$ L(S_p^n,t) \le (t/\kappa_{p}(A))^{(c/b)^2} = (Ct)^{(c/b)^2} \to 0. $$

Let $X$ be a random variable on $\mathbb R^n$ and let $S_p^n := \{w \in \mathbb R^n \mid \|w\|_p = 1\}$ be the unit-sphere w.r.t to the $\ell_p$-norm in $\mathbb R^n$. We will be particularly interested in the case $p \in \{1,2\}$. For any $w \in B_p^n$ and $t \ge 0$, let $$ L(X'w,t):= \sup_{u \in \mathbb R}\mathbb P(|X' w - u| \le t). $$ be the Lévy concentration function of the random variable $X' w$. Finally, define $$ L(S_p^n,t) := \sup_{w \in S_p^n} L(X'w,t). $$

Question. Under what minimal assumptions on $X$ is it true that $\lim_{t \to 0^+} L(S_p^n, t) \to 0 $ ?

For example, does it suffice to assume that

• $X$ has density which is sufficiently smooth (e.g continuous) ?
• What if we simply assume that the distribution of $X$ is atomless ?

Some solved cases

• (1) Bounded Radon transform of density of $X$. Suppose there exists a positive constant $b$ such that for every $w \in B_p^n$, the random variable $X'w$ has density bounded by $b$. Then, $$ L(S_p^n,t) = \sup_{w \in S_p^n}\sup_{u \in \mathbb R}\mathbb P(|X'w-u| \le t) \le b\cdot 2t = 2bt \to 0. $$ For example, this is the case when the distribution of $X$ is a mixture of multivariate Gaussians $\sum_k\pi_kN(\mu_k,\Sigma_k)$; in this case each $X'w$ has distribution $\sum_k \pi_k N(\mu_k'w,w'\Sigma_k w)$, and the condition clearly holds with $b=1$. Finally, note that if $f$ is the density of $X$, then the density $\rho_w$ of $X'w$ evaluated at a point $c$ is nothing but the Radon transform $R[f]$ of $f$ w.r.t the hyperplane $H_{w,c} := \{x \in \mathbb R^n \mid x'w = c\}$. Thus, the assumption that $\sup_{w \in S_p^n}\|\rho_w\|_\infty \le b$, is really reminiscent of demanding that $R[f] \in L^\infty$. So the question now is, under what minimal conditions if the Radon transform of a density function bounded ?

• (2) $X$ is an transformation of log-concave random vector. C

onsider the scenario where $X \overset{D}{=}AZ + \mu$, where $A$ is a deterministic $m \times n$ matrix, $\mu$ is a determistic vector in $\mathbb R^m$, and $Z$ is random variable which is isotropic, log-concave, whose coordinates are $b$-subGaussian, then thanks to Grigoris Paouris' Small Ball Probability Estimates for Log-Concave Measures, we know that $$ L(X'w,t) = L((Aw)'X,t) \le (t/\|Aw\|_2)^{(c/b)^2},\text{ for sufficiently small }t $$ where $c$ is an absolute positive constant. Thus, if $A$ is non-degenerate in the sense that $\kappa_p(A) := \inf_{w \in S_p^n} \|Aw\|_2 \ge 1/C > 0$, then

$$ L(S_p^n,t) \le (t/\kappa_{p}(A))^{(c/b)^2} = (Ct)^{(c/b)^2} \to 0. $$

Let $X$ be a random variable on $\mathbb R^n$ and let $S_p^n := \{w \in \mathbb R^n \mid \|w\|_p = 1\}$ be the unit-sphere w.r.t to the $\ell_p$-norm in $\mathbb R^n$. We will be particularly interested in the case $p \in \{1,2\}$. For any $w \in B_p^n$ and $t \ge 0$, let $$ L(X'w,t):= \sup_{u \in \mathbb R}\mathbb P(|X' w - u| \le t). $$ be the Lévy concentration function of the random variable $X' w$. Finally, define $$ L(S_p^n,t) := \sup_{w \in S_p^n} L(X'w,t). $$

Question. Under what minimal assumptions on $X$ is it true that $\lim_{t \to 0^+} L(S_p^n, t) \to 0 $ ?

For example, does it suffice to assume that

• $X$ has density which is sufficiently smooth (e.g continuous) ?
• What if we simply assume that the distribution of $X$ is atomless ?

Some solved cases

• (1) Bounded Radon transform of density of $X$. Suppose there exists a positive constant $b$ such that for every $w \in B_p^n$, the random variable $X'w$ has density bounded by $b$. Then, $$ L(S_p^n,t) = \sup_{w \in S_p^n}\sup_{u \in \mathbb R}\mathbb P(|X'w-u| \le t) \le b\cdot 2t = 2bt \to 0. $$ For example, this is the case when the distribution of $X$ is a mixture of multivariate Gaussians $\sum_k\pi_kN(\mu_k,\Sigma_k)$; in this case each $X'w$ has distribution $\sum_k \pi_k N(\mu_k'w,w'\Sigma_k w)$, and the condition clearly holds with $b=1$. Finally, note that if $f$ is the density of $X$, then the density $\rho_w$ of $X'w$ evaluated at a point $c$ is nothing but the Radon transform $R[f]$ of $f$ w.r.t the hyperplane $H_{w,c} := \{x \in \mathbb R^n \mid x'w = c\}$. Thus, the assumption that $\sup_{w \in S_p^n}\|\rho_w\|_\infty \le b$, is really reminiscent of demanding that $R[f] \in L^\infty$. So the question now is, under what minimal conditions if the Radon transform of a density function bounded ?

• (2) $X$ admits a vanishing Small-ball probability. Let us now show that $L(S_p^n,t) \to 0$ if $X$ admits small-ball probability which vanishes at zero, i.e $$ \tag{1} L(X,t) := \sup_{x \in \mathbb R^n}\mathbb P(\|X-x\|_2 \le t) \to 0\text{ in the limit }t \to 0^+. $$ In particular, this is the case when $X$ has bounded density $f$, since $$ L(X,t) \le \|f\|_\infty\cdot \sup_{x\in \mathbb R^n}\mbox{vol}(tB_2^n + x) \le C_n\cdot t^n \to 0. $$ Now, define $\kappa_{n,p} := \inf\{\|w\|_2 \mid w \in B_p^n\} > 0$. It is easy to see that $\kappa_{n,p} = 1/n^{(1/p-1/2)_+}$. For example, $\kappa_{n,1} = 1/\sqrt{n}$ and $\kappa_{n,2} = 1$. Under the condition (1), one then computes $$ \begin{split} L(S_p^n,t) &= \sup_{w \in S_p^n}\sup_{u \in \mathbb R}\mathbb P(|X'w-u| \le t) \le \sup_{w \in S_p^n}\sup_{x \in \mathbb R^n}\mathbb P(|X'w-x'w| \le t)\\ & \le \sup_{w \in S_p^n}\sup_{x \in \mathbb R^n}\mathbb P(\|w\|_2\|X-x\|_2 \le t)\\ &\le \sup_{x \in \mathbb R^n}\mathbb P(\|X-x\|_2 \le t/\kappa_{n,p}),\text{ since }\|w\|_2 \ge \kappa_{n,p}\text{ if }\|w\|_p = 1\\ &= L(X,t/\kappa_{n,p}) \to 0,\text{ by (1)} \end{split} $$

• (3) $X$ is an transformation of log-concave random vector. As a special example of the preceding case, consider the scenario where $X \overset{D}{=}AZ + \mu$, where $A$ is a deterministic $m \times n$ matrix, $\mu$ is a determistic vector in $\mathbb R^m$, and $Z$ is random variable which is isotropic, log-concave, whose coordinates are $b$-subGaussian, then thanks to Grigoris Paouris' Small Ball Probability Estimates for Log-Concave Measures, we know that $$ L(X,t) \le (t/\|A\|_F)^{(c/b)^2r(A)},\text{ for sufficiently small }t $$ where $c$ is an absolute positive constant and $r(A) := \|A\|_F^2 / \|A\|_{op}^2$ is the effective rank of $A$. We conclude that $L(B_p^n,t) \to 0$ in the limit $t \to 0^+$.

Let $X$ be a random variable on $\mathbb R^n$ and let $S_p^n := \{w \in \mathbb R^n \mid \|w\|_p = 1\}$ be the unit-sphere w.r.t to the $\ell_p$-norm in $\mathbb R^n$. We will be particularly interested in the case $p \in \{1,2\}$. For any $w \in B_p^n$ and $t \ge 0$, let $$ L(X'w,t):= \sup_{u \in \mathbb R}\mathbb P(|X' w - u| \le t). $$ be the Lévy concentration function of the random variable $X' w$. Finally, define $$ L(S_p^n,t) := \sup_{w \in S_p^n} L(X'w,t). $$

Question. Under what minimal assumptions on $X$ is it true that $\lim_{t \to 0^+} L(S_p^n, t) \to 0 $ ?

For example, does it suffice to assume that

• $X$ has density which is sufficiently smooth (e.g continuous) ?
• What if we simply assume that the distribution of $X$ is atomless ?

Some solved cases

• (1) Bounded Radon transform of density of $X$. Suppose there exists a positive constant $b$ such that for every $w \in B_p^n$, the random variable $X'w$ has density bounded by $b$. Then, $$ L(S_p^n,t) = \sup_{w \in S_p^n}\sup_{u \in \mathbb R}\mathbb P(|X'w-u| \le t) \le b\cdot 2t = 2bt \to 0. $$ For example, this is the case when the distribution of $X$ is a mixture of multivariate Gaussians $\sum_k\pi_kN(\mu_k,\Sigma_k)$; in this case each $X'w$ has distribution $\sum_k \pi_k N(\mu_k'w,w'\Sigma_k w)$, and the condition clearly holds with $b=1$. Finally, note that if $f$ is the density of $X$, then the density $\rho_w$ of $X'w$ evaluated at a point $c$ is nothing but the Radon transform $R[f]$ of $f$ w.r.t the hyperplane $H_{w,c} := \{x \in \mathbb R^n \mid x'w = c\}$. Thus, the assumption that $\sup_{w \in S_p^n}\|\rho_w\|_\infty \le b$, is really reminiscent of demanding that $R[f] \in L^\infty$. So the question now is, under what minimal conditions if the Radon transform of a density function bounded ?

• (2) $X$ admits a vanishing Small-ball probability. Let us now show that $L(S_p^n,t) \to 0$ if $X$ admits small-ball probability which vanishes at zero, i.e $$ \tag{1} L(X,t) := \sup_{x \in \mathbb R^n}\mathbb P(\|X-x\|_2 \le t) \to 0\text{ in the limit }t \to 0^+. $$ In particular, this is the case when $X$ has bounded density $f$, since $$ L(X,t) \le \|f\|_\infty\cdot \sup_{x\in \mathbb R^n}\mbox{vol}(tB_2^n + x) \le C_n\cdot t^n \to 0. $$ Now, define $\kappa_{n,p} := \inf\{\|w\|_2 \mid w \in B_p^n\} > 0$. It is easy to see that $\kappa_{n,p} = 1/n^{(1/p-1/2)_+}$. For example, $\kappa_{n,1} = 1/\sqrt{n}$ and $\kappa_{n,2} = 1$. Under the condition (1), one then computes $$ \begin{split} L(S_p^n,t) &= \sup_{w \in S_p^n}\sup_{u \in \mathbb R}\mathbb P(|X'w-u| \le t) \le \sup_{w \in S_p^n}\sup_{x \in \mathbb R^n}\mathbb P(|X'w-x'w| \le t)\\ & \le \sup_{w \in S_p^n}\sup_{x \in \mathbb R^n}\mathbb P(\|w\|_2\|X-x\|_2 \le t)\\ &\le \sup_{x \in \mathbb R^n}\mathbb P(\|X-x\|_2 \le t/\kappa_{n,p}),\text{ since }\|w\|_2 \ge \kappa_{n,p}\text{ if }\|w\|_p = 1\\ &= L(X,t/\kappa_{n,p}) \to 0,\text{ by (1)} \end{split} $$

• (3) $X$ is an transformation of log-concave random vector. C

onsider the scenario where $X \overset{D}{=}AZ + \mu$, where $A$ is a deterministic $m \times n$ matrix, $\mu$ is a determistic vector in $\mathbb R^m$, and $Z$ is random variable which is isotropic, log-concave, whose coordinates are $b$-subGaussian, then thanks to Grigoris Paouris' Small Ball Probability Estimates for Log-Concave Measures, we know that $$ L(X'w,t) = L((Aw)'X,t) \le (t/\|Aw\|_2)^{(c/b)^2},\text{ for sufficiently small }t $$ where $c$ is an absolute positive constant. Thus, if $A$ is non-degenerate in the sense that $\kappa_p(A) := \inf_{w \in S_p^n} \|Aw\|_2 \ge 1/C > 0$, then

$$ L(S_p^n,t) \le (t/\kappa_{p}(A))^{(c/b)^2} = (Ct)^{(c/b)^2} \to 0. $$

Let $X$ be a random variable on $\mathbb R^n$ and let $S_p^n := \{w \in \mathbb R^n \mid \|w\|_p = 1\}$ be the unit-sphere w.r.t to the $\ell_p$-norm in $\mathbb R^n$. We will be particularly interested in the case $p \in \{1,2\}$. For any $w \in B_p^n$ and $t \ge 0$, let $$ L(X'w,t):= \sup_{u \in \mathbb R}\mathbb P(|X' w - u| \le t). $$ be the Lévy concentration function of the random variable $X' w$. Finally, define $$ L(S_p^n,t) := \sup_{w \in S_p^n} L(X'w,t). $$

Question. Under what minimal assumptions on $X$ is it true that $\lim_{t \to 0^+} L(S_p^n, t) \to 0 $ ?

For example, does it suffice to assume that $X$ has sufficiently smooth density on $\mathbb R^n$ ?

Some solved cases

• (1) Bounded Radon transform of density of $X$. Suppose there exists a positive constant $b$ such that for every $w \in B_p^n$, the random variable $X'w$ has density bounded by $b$. Then, $$ L(S_p^n,t) = \sup_{w \in S_p^n}\sup_{u \in \mathbb R}\mathbb P(|X'w-u| \le t) \le b\cdot 2t = 2bt \to 0. $$ For example, this is the case when the distribution of $X$ is a mixture of multivariate Gaussians $\sum_k\pi_kN(\mu_k,\Sigma_k)$; in this case each $X'w$ has distribution $\sum_k \pi_k N(\mu_k'w,w'\Sigma_k w)$, and the condition clearly holds with $b=1$. Finally, note that if $f$ is the density of $X$, then the density $\rho_w$ of $X'w$ evaluated at a point $c$ is nothing but the Radon transform $R[f]$ of $f$ w.r.t the hyperplane $H_{w,c} := \{x \in \mathbb R^n \mid x'w = c\}$. Thus, the assumption that $\sup_{w \in S_p^n}\|\rho_w\|_\infty \le b$, is really reminiscent of demanding that $R[f] \in L^\infty$. So the question now is, under what minimal conditions if the Radon transform of a density function bounded ?

• (2) $X$ admits a vanishing Small-ball probability. Let us now show that $L(S_p^n,t) \to 0$ if $X$ admits small-ball probability which vanishes at zero, i.e $$ \tag{1} L(X,t) := \sup_{x \in \mathbb R^n}\mathbb P(\|X-x\|_2 \le t) \to 0\text{ in the limit }t \to 0^+. $$ In particular, this is the case when $X$ has bounded density $f$, since $$ L(X,t) \le \|f\|_\infty\cdot \sup_{x\in \mathbb R^n}\mbox{vol}(tB_2^n + x) \le C_n\cdot t^n \to 0. $$ Now, define $\kappa_{n,p} := \inf\{\|w\|_2 \mid w \in B_p^n\} > 0$. It is easy to see that $\kappa_{n,p} = 1/n^{(1/p-1/2)_+}$. For example, $\kappa_{n,1} = 1/\sqrt{n}$ and $\kappa_{n,2} = 1$. Under the condition (1), one then computes $$ \begin{split} L(S_p^n,t) &= \sup_{w \in S_p^n}\sup_{u \in \mathbb R}\mathbb P(|X'w-u| \le t) \le \sup_{w \in S_p^n}\sup_{x \in \mathbb R^n}\mathbb P(|X'w-x'w| \le t)\\ & \le \sup_{w \in S_p^n}\sup_{x \in \mathbb R^n}\mathbb P(\|w\|_2\|X-x\|_2 \le t)\\ &\le \sup_{x \in \mathbb R^n}\mathbb P(\|X-x\|_2 \le t/\kappa_{n,p}),\text{ since }\|w\|_2 \ge \kappa_{n,p}\text{ if }\|w\|_p = 1\\ &= L(X,t/\kappa_{n,p}) \to 0,\text{ by (1)} \end{split} $$

• (3) $X$ is an transformation of log-concave random vector. As a special example of the preceding case, consider the scenario where $X \overset{D}{=}AZ + \mu$, where $A$ is a deterministic $m \times n$ matrix, $\mu$ is a determistic vector in $\mathbb R^m$, and $Z$ is random variable which is isotropic, log-concave, whose coordinates are $b$-subGaussian, then thanks to Grigoris Paouris' Small Ball Probability Estimates for Log-Concave Measures, we know that $$ L(X,t) \le (t/\|A\|_F)^{(c/b)^2r(A)},\text{ for sufficiently small }t $$ where $c$ is an absolute positive constant and $r(A) := \|A\|_F^2 / \|A\|_{op}^2$ is the effective rank of $A$. We conclude that $L(B_p^n,t) \to 0$ in the limit $t \to 0^+$.

Let $X$ be a random variable on $\mathbb R^n$ and let $S_p^n := \{w \in \mathbb R^n \mid \|w\|_p = 1\}$ be the unit-sphere w.r.t to the $\ell_p$-norm in $\mathbb R^n$. We will be particularly interested in the case $p \in \{1,2\}$. For any $w \in B_p^n$ and $t \ge 0$, let $$ L(X'w,t):= \sup_{u \in \mathbb R}\mathbb P(|X' w - u| \le t). $$ be the Lévy concentration function of the random variable $X' w$. Finally, define $$ L(S_p^n,t) := \sup_{w \in S_p^n} L(X'w,t). $$

Question. Under what minimal assumptions on $X$ is it true that $\lim_{t \to 0^+} L(S_p^n, t) \to 0 $ ?

For example, does it suffice to assume that

• $X$ has density which is sufficiently smooth (e.g continuous) ?
• What if we simply assume that the distribution of $X$ is atomless ?

Some solved cases

• (1) Bounded Radon transform of density of $X$. Suppose there exists a positive constant $b$ such that for every $w \in B_p^n$, the random variable $X'w$ has density bounded by $b$. Then, $$ L(S_p^n,t) = \sup_{w \in S_p^n}\sup_{u \in \mathbb R}\mathbb P(|X'w-u| \le t) \le b\cdot 2t = 2bt \to 0. $$ For example, this is the case when the distribution of $X$ is a mixture of multivariate Gaussians $\sum_k\pi_kN(\mu_k,\Sigma_k)$; in this case each $X'w$ has distribution $\sum_k \pi_k N(\mu_k'w,w'\Sigma_k w)$, and the condition clearly holds with $b=1$. Finally, note that if $f$ is the density of $X$, then the density $\rho_w$ of $X'w$ evaluated at a point $c$ is nothing but the Radon transform $R[f]$ of $f$ w.r.t the hyperplane $H_{w,c} := \{x \in \mathbb R^n \mid x'w = c\}$. Thus, the assumption that $\sup_{w \in S_p^n}\|\rho_w\|_\infty \le b$, is really reminiscent of demanding that $R[f] \in L^\infty$. So the question now is, under what minimal conditions if the Radon transform of a density function bounded ?

• (2) $X$ admits a vanishing Small-ball probability. Let us now show that $L(S_p^n,t) \to 0$ if $X$ admits small-ball probability which vanishes at zero, i.e $$ \tag{1} L(X,t) := \sup_{x \in \mathbb R^n}\mathbb P(\|X-x\|_2 \le t) \to 0\text{ in the limit }t \to 0^+. $$ In particular, this is the case when $X$ has bounded density $f$, since $$ L(X,t) \le \|f\|_\infty\cdot \sup_{x\in \mathbb R^n}\mbox{vol}(tB_2^n + x) \le C_n\cdot t^n \to 0. $$ Now, define $\kappa_{n,p} := \inf\{\|w\|_2 \mid w \in B_p^n\} > 0$. It is easy to see that $\kappa_{n,p} = 1/n^{(1/p-1/2)_+}$. For example, $\kappa_{n,1} = 1/\sqrt{n}$ and $\kappa_{n,2} = 1$. Under the condition (1), one then computes $$ \begin{split} L(S_p^n,t) &= \sup_{w \in S_p^n}\sup_{u \in \mathbb R}\mathbb P(|X'w-u| \le t) \le \sup_{w \in S_p^n}\sup_{x \in \mathbb R^n}\mathbb P(|X'w-x'w| \le t)\\ & \le \sup_{w \in S_p^n}\sup_{x \in \mathbb R^n}\mathbb P(\|w\|_2\|X-x\|_2 \le t)\\ &\le \sup_{x \in \mathbb R^n}\mathbb P(\|X-x\|_2 \le t/\kappa_{n,p}),\text{ since }\|w\|_2 \ge \kappa_{n,p}\text{ if }\|w\|_p = 1\\ &= L(X,t/\kappa_{n,p}) \to 0,\text{ by (1)} \end{split} $$

• (3) $X$ is an transformation of log-concave random vector. As a special example of the preceding case, consider the scenario where $X \overset{D}{=}AZ + \mu$, where $A$ is a deterministic $m \times n$ matrix, $\mu$ is a determistic vector in $\mathbb R^m$, and $Z$ is random variable which is isotropic, log-concave, whose coordinates are $b$-subGaussian, then thanks to Grigoris Paouris' Small Ball Probability Estimates for Log-Concave Measures, we know that $$ L(X,t) \le (t/\|A\|_F)^{(c/b)^2r(A)},\text{ for sufficiently small }t $$ where $c$ is an absolute positive constant and $r(A) := \|A\|_F^2 / \|A\|_{op}^2$ is the effective rank of $A$. We conclude that $L(B_p^n,t) \to 0$ in the limit $t \to 0^+$.