Consider a convex compact set $K\subset R^n$ (with non-empty interior if that helps).

Let $\Pi_K:R^n\to K$ be the projection onto $K$, defined as $$ \Pi_K(x) = \text{argmin}_{k\in K} \|k-x\| $$ where, throughout, $\|\cdot\|$ is the Euclidean norm.

In a problem in convex geometry (random linear transform of $K$), I am encountering the quantities $$ \inf_{t>0} \frac{E[\|tg - \Pi_K(t g)\|]}{t}, \qquad \inf_{t>0} \frac{E[\|tg - \Pi_K(t g)\|^2]^{1/2}}{t} $$ where $\|\Pi_K(t g)\|$ is the Euclidean norm of the projection of $tg$ onto $K$. Above, $g\in N(0,I_n)$ and the expectation is with respect to the distribution of $g$. It would still be of interest if $g$ were replaced by a uniform random vector on the sphere $S^{n-1}$.

If $K$ were a subspace, these quantities would capture the dimension. For a cone $K$ pointed at 0, these quantities reduce to the statistical dimension studied in [1]. I am wondering if these quantities have been studied, if they are known to have equivalent definitions related to other properties of $K$.

[1]: Dennis Amelunxen, Martin Lotz, Michael B. McCoy, Joel A. Tropp, Living on the edge: phase transitions in convex programs with random data, Information and Inference: A Journal of the IMA, Volume 3, Issue 3, September 2014, Pages 224–294, https://doi.org/10.1093/imaiai/iau005

Consider a convex compact set $K\subset R^n$ (with non-empty interior if that helps).

Let $\Pi_K:R^n\to K$ be the projection onto $K$, defined as $$ \Pi_K(x) = \operatorname{argmin}_{k\in K} \|k-x\| $$ where, throughout, $\|\cdot\|$ is the Euclidean norm.

In a problem in convex geometry (random linear transform of $K$), I am encountering the quantities $$ \inf_{t>0} \frac{E[\|tg - \Pi_K(t g)\|]}{t}, \qquad \inf_{t>0} \frac{E[\|tg - \Pi_K(t g)\|^2]^{1/2}}{t}. $$ Above, $g\in N(0,I_n)$ and the expectation is with respect to the distribution of $g$. It would still be of interest if $g$ were replaced by a uniform random vector on the sphere $S^{n-1}$.

If $K$ were a subspace, these quantities would capture the dimension. For a cone $K$ pointed at 0, these quantities reduce to the statistical dimension studied in [1]. I am wondering if these quantities have been studied, if they are known to have equivalent definitions related to other properties of $K$.

[1]: Dennis Amelunxen, Martin Lotz, Michael B. McCoy, Joel A. Tropp, Living on the edge: phase transitions in convex programs with random data, Information and Inference: A Journal of the IMA, Volume 3, Issue 3, September 2014, Pages 224–294, https://doi.org/10.1093/imaiai/iau005

Consider a convex compact set $K\subset R^n$ (with non-empty interior if that helps).

Let $\Pi_K:R^n\to K$ be the projection onto $K$, defined as $$ \Pi_K(x) = \text{argmin}_{k\in K} \|k-x\| $$ where, throughout, $\|\cdot\|$ is the Euclidean norm.

In a problem in convex geometry (random linear transform of $K$), I am encountering the quantities $$ \inf_{t>0} \frac{E[\|\Pi_K(t g)\|]}{t}, \qquad \inf_{t>0} \frac{E[\|\Pi_K(t g)\|^2]^{1/2}}{t} $$ where $\|\Pi_K(t g)\|$ is the Euclidean norm of the projection of $tg$ onto $K$. Above, $g\in N(0,I_n)$ and the expectation is with respect to the distribution of $g$. It would still be of interest if $g$ were replaced by a uniform random vector on the sphere $S^{n-1}$.

If $K$ were a subspace, these quantities would capture the dimension. For a cone $K$ pointed at 0, these quantities reduce to the statistical dimension studied in [1]. I am wondering if these quantities have been studied, if they are known to have equivalent definitions related to other properties of $K$.

[1]: Dennis Amelunxen, Martin Lotz, Michael B. McCoy, Joel A. Tropp, Living on the edge: phase transitions in convex programs with random data, Information and Inference: A Journal of the IMA, Volume 3, Issue 3, September 2014, Pages 224–294, https://doi.org/10.1093/imaiai/iau005

Consider a convex compact set $K\subset R^n$ (with non-empty interior if that helps).

Let $\Pi_K:R^n\to K$ be the projection onto $K$, defined as $$ \Pi_K(x) = \text{argmin}_{k\in K} \|k-x\| $$ where, throughout, $\|\cdot\|$ is the Euclidean norm.

In a problem in convex geometry (random linear transform of $K$), I am encountering the quantities $$ \inf_{t>0} \frac{E[\|tg - \Pi_K(t g)\|]}{t}, \qquad \inf_{t>0} \frac{E[\|tg - \Pi_K(t g)\|^2]^{1/2}}{t} $$ where $\|\Pi_K(t g)\|$ is the Euclidean norm of the projection of $tg$ onto $K$. Above, $g\in N(0,I_n)$ and the expectation is with respect to the distribution of $g$. It would still be of interest if $g$ were replaced by a uniform random vector on the sphere $S^{n-1}$.

If $K$ were a subspace, these quantities would capture the dimension. For a cone $K$ pointed at 0, these quantities reduce to the statistical dimension studied in [1]. I am wondering if these quantities have been studied, if they are known to have equivalent definitions related to other properties of $K$.

[1]: Dennis Amelunxen, Martin Lotz, Michael B. McCoy, Joel A. Tropp, Living on the edge: phase transitions in convex programs with random data, Information and Inference: A Journal of the IMA, Volume 3, Issue 3, September 2014, Pages 224–294, https://doi.org/10.1093/imaiai/iau005

Consider a convex compact set $K\subset R^n$ (with non-empty interior if that helps).

Let $\Pi_K:R^n\to K$ be the convex projection onto $K$, defined as $$ \Pi_K(x) = \text{argmin}_{k\in K} \|k-x\| $$ where, throughout, $\|\cdot\|$ is the Euclidean norm.

In a problem in convex geometry (random linear transform of $K$), I am encountering the quantities $$ \inf_{t>0} \frac{E[\|\Pi_K(t g)\|]}{t}, \qquad \inf_{t>0} \frac{E[\|\Pi_K(t g)\|^2]^{1/2}}{t} $$ where $\|\Pi_K(t g)\|$ is the Euclidean norm of the convex projection of $tg$ onto $K$. Above, $g\in N(0,I_n)$ and the expectation is with respect to the distribution of $g$. It would still be of interest if $g$ were replaced by a uniform random vector on the sphere $S^{n-1}$.

If $K$ were a subspace, these quantities would capture the dimension. For a cone $K$ pointed at 0, these quantities reduce to the statistical dimension studied in [1]. I am wondering if these quantities have been studied, if they are known to have equivalent definitions related to other properties of $K$.

[1]: Dennis Amelunxen, Martin Lotz, Michael B. McCoy, Joel A. Tropp, Living on the edge: phase transitions in convex programs with random data, Information and Inference: A Journal of the IMA, Volume 3, Issue 3, September 2014, Pages 224–294, https://doi.org/10.1093/imaiai/iau005

Consider a convex compact set $K\subset R^n$ (with non-empty interior if that helps).

Let $\Pi_K:R^n\to K$ be the projection onto $K$, defined as $$ \Pi_K(x) = \text{argmin}_{k\in K} \|k-x\| $$ where, throughout, $\|\cdot\|$ is the Euclidean norm.

In a problem in convex geometry (random linear transform of $K$), I am encountering the quantities $$ \inf_{t>0} \frac{E[\|\Pi_K(t g)\|]}{t}, \qquad \inf_{t>0} \frac{E[\|\Pi_K(t g)\|^2]^{1/2}}{t} $$ where $\|\Pi_K(t g)\|$ is the Euclidean norm of the projection of $tg$ onto $K$. Above, $g\in N(0,I_n)$ and the expectation is with respect to the distribution of $g$. It would still be of interest if $g$ were replaced by a uniform random vector on the sphere $S^{n-1}$.

If $K$ were a subspace, these quantities would capture the dimension. For a cone $K$ pointed at 0, these quantities reduce to the statistical dimension studied in [1]. I am wondering if these quantities have been studied, if they are known to have equivalent definitions related to other properties of $K$.

[1]: Dennis Amelunxen, Martin Lotz, Michael B. McCoy, Joel A. Tropp, Living on the edge: phase transitions in convex programs with random data, Information and Inference: A Journal of the IMA, Volume 3, Issue 3, September 2014, Pages 224–294, https://doi.org/10.1093/imaiai/iau005