# Machine Learning Based Nitrate Concentration Soft-Sensor

Harry Dobbs

(December 2022)

Author | Harry Dobbs
---|---

## Summary

## Introduction

Excessive amounts of nitrogen in rivers is a major concern worldwide. The
increased amount of nitrogen in rivers is a concern for aquatic life, drinking
water and recreational activities. In New Zealand, the increase in nitrates is
primarily caused by intensification of agriculture [4]. To enable mitigation
of the effects caused by nitrate leeching, monitoring of nitrate levels is
required. Traditionally this has been achieved by river grab sampling which
has numerous drawbacks like delays in data acquisition, high costs and
difficulty to detect events that take place during the time interval between
two samples [9]. New Zealand has recognized that monthly discrete (spot) water
quality measurements cannot capture or characterize rapid changes in water
quality. High frequency nitrate monitoring is becoming popular important for
several reasons, such as improving uncertainty in the examination of trends in
nitrate concentrations, improving accuracy of estimated nitrate loads,
calibration of surface water and ground water nitrate transport models,
improving ability to predict downstream water quality and identifying periods
when nitrate concentrations require additional management [10].

The common commercially available sensor technologies can be categorized as
ion-selective electrodes (ISE), wet-chemical analysers and optical sensors.
ISE is relatively inexpensive and easy to use, however they are often
inaccurate and prone to drifting. Wet chemical and optical sensors are
considered to be higher accuracy but are expensive and require significant
maintenance [1]. These issues hinder the possibility of nitrate sensors being
able to perform high-frequency in-situ nitrate monitoring. Recently soft-
sensor machine learning based approaches have been explored. Soft-sensor based
approaches use available surrogate data such as water temperature,
conductivity, turbidity, dissolved oxygen and pH to determine target variables
such as nitrate or phosphorus concentrations. Soft-sensor based approaches can
be considered low cost and low maintenance. However, their accuracy
significantly relies on the ability to have an appropriate model - which
generally requires large quantities of ground truth data to train and validate
the model.

The work presented in this paper aims to aid in the development of a robust,
high frequency, low cost and high accuracy nitrate concentration sampling
methodology. The main objectives of this work are as follows:

  * •

Report on the current state-of-art nitrate concentration soft-sensors.

  * •

Determine which machine learning model would be most suitable for real-time
nitrate monitoring (considering both non-linear and linear models).

  * •

Further understand which predictor variables are most beneficial to the
nitrate concentration model performance.

  * •

Quantify how much training data is required to train a model that can work on
real-time data with acceptable accuracy for long-term stable deployement.

## Related Work

The most popular models used for predicting nitrate concentrations from
surrogate variables include random forests (RF), support vector machines
(SVM), Linear Regression (LR), neural networks (NN) and Gaussian mixture
models (GAMMS).

Random Forest (RF): Random forests regression is a powerful and widely used
supervised machine learning method - capable of modelling non-linear patterns.
RF builds a multitude of regression trees (forest) by making many random
subsets of the data and using random variable sets to build many predictions
of the dependent variable. Those many predictions are averaged to produce the
final estimated dependent variable. A number of high frequency nitrate soft-
sensor implementations have used random forest regression [5, 7, 12, 1].

Green et al [5] implements a RF and SVM model using high frequency data (15
minute intervals) from four years (October 2012 - January 2017) worth of data
collected from the Hubbard Brooke Experimental Forest River based in the White
Mountains of New Hampshire, USA. The catchment is a steep (mean slope = 17.1%)
glaciated hillside of spodic soils. The sensor data collected includes EC, pH,
fDOM, turbidity, DO, NO3. Additionally discrete routine weekly samples and
multiple storm and seasonal snow melt sampling campaigns were taken. To
incorporate seasonality as a predictor variable, day of the year was encoded.
The models were trained using the manual samples (n=382) which then allows the
models to predict nitrate concentrations at 15-minute intervals. The results
from this study showed great promise for nitrate concentration prediction when
an appropriate sensor or proxy is not available (Nash-Sutcliffe efficiencies;
$R^{2}$ = 0.68 and 0.64 for the SVM and RF models, respectively).

The results achieved by Green et al show the SVM model slightly out preforms
the RF model (RF preformed slightly better for Aluminium and Potassium
concentration). However the RF model was found to be more robust when
predictor variables were removed. This paper mentions how additional
independent variables such as meteorological conditions (temperature,
precipitation, vapour pressure deficit, evapo-transpiration) could aid in
better predictions. Future work recommended by this paper was to build a
number of models to represent a range of catchments, then test the ability for
the models to generalize across time and place (could not be achieved in this
paper due to lack of available data).

Harrison et al [7], proposed a solution to estimate stream nitrogen and
phosphorus concentrations from sensor data from 11 locations within a
forested, mountainous drainage area in upstate New York by using a RF model.
The catchment is predominantly forested, though parts of the lake’s shoreline
are relatively well-developed. The high frequency sensor data used in this
study includes hydro-static pressure (HP), soil moisture (SM), water
temperature, pH, specific conductance, turbidity, fluorescence dissolved
organic matter (fDOM). The model was trained using these sensors as well as
their 1-h, 5-h, 24-h, and 120-h lags and delta values. The manual water
nutrient samples were taken monthly, and the sensor values were captured every
1-15 minutes. During storm events, water nutrients samples were taken
automatically at a rate of 1-2hrs. The manual water nutrient samples were
paired with the high frequency data hourly median values. Leave one out cross
validation was used to train and test this method (LOOCV). LOOCV involves
omitting one case (row) at a time from the dataset while training (fitting)
the model, then predicting the value for the omitted case. This is done n
times, where n is the number of values of the dependent variable. This method
was modified slightly, omitting data from one sampling event at a time (i.e.,
1 case from baseflow dates, multiple cases from storm event dates when
multiple observations were made on the same day). The results from this study
were (MAE=0.075mgN/l, RMSE=0.12mgN/l, R2=0.89) for detecting nitrates where
the nitrate ranges were from 0.01-1.01 mg/L .

This study found fDOM and soil moisture to be important predictors for
calculating nutrient concentrations. The authors noted a disadvantage of
random forests is their inability to make predictions outside of the data used
for training. However this problem exists in all machine learning models
(inability to accurately extrapolate outside the training values).

Jingshui et al [12] also developed a RF model to determine nitrates using high
frequency predictor data (15 minute intervals) from three locations on the
Danube River, Germany. The relevant predictor variables used include dissolved
oxygen, temperature, conductivity, pH, and discharge rate (Q). Monthly, daily
and hourly features were also implemented into the model. The RF results were
compared against a multi-variable linear regression model. The RF demonstrated
better results when using 3 or more predictor variables. This study implements
a moving mean filter to remove outliers by using a sliding windows of 6h (24
data points). If a value is more of less than three local std deviations from
the local mean the value will be defined as an outlier.

Similarly Castrillo et al [1] implemented a RF model to determine nitrate
concentrations from two different monitoring sites from contributors to the
River Thames England (Enborne and The Cut). The River Enborne is relatively
rural, receiving sewage effluent from small sewage treatment works, whereas
The Cut is highly urbanised, receiving large quantities of sewage from major
towns. This study implements a preprocessing stage to help remove data skew,
prior to creating training, testing and validation data-sets. This method
similar to Jingshui et al[12], creates the data-sets using a K-fold sampling
scheme (K=10) , which results in 90% of the data being used as training data.
The model to achieve impress results (nRMSE of 15% and RMSE 0.025 mg/l on the
Enborne data-set), with EC and pH being the most powerful variables. The work
in this study shows that RF is an appropriate model choice and are able to
effectively represent complex environmental phenomena.

Recently Paepae et al [11] expanded on the work done by Castrillo by testing
different machine learning models, data imputation methods for missing values
(and data scaling methods) and proposing the best practises for soft-sensor
implementations. The authors evaluated decision tree (DT), random forest (RF),
extremegradient boosting (XGB), light gradient boosting machine (LGBM), multi-
layer perceptron (MLP), k-nearest neighbor (kNN), support vector machine
(SVM), extremely randomized trees (ET), gradient boosting regressor (GB),
stochastic gradient descent (SGD), linear regression (LR), ridge regression
(Ridge), and bagging regressor (BR). It was observed that ET preformed the
best (R2=0.9682), followed by RF (R2=0.9544) and LR preformed the worst.
Little to no-benefit was found by scaling the values. The effects of value
imputation and pre-transformations were found to be marginal.

Neural Networks: Neural networks are another popular methods for determining
Nitrate concentrations from surrogate sensors and predictors. Neural networks
work by learning a function embedding which is encoded by connecting
artificial neurons through sequential layers.

An neural network was proposed by Diamantopoulou et al [3] for the prediction
of the monthly values of three water quality parameters of the Strymon river
at a station located in Sidirokastro Bridge near the Greek - Bulgarian borders
by using the monthly values of the other existing water quality parameters as
input variables. The monthly data of thirteen parameters and the discharge, at
the Sidirokastro station, for the time period 1980-1990 were selected for this
analysis. The results demonstrate the ability of the appropriate ANN models
for the prediction of water quality parameters. This provides a very useful
tool for filling the missing values that is a very serious problem in most of
the Greek monitoring stations.

A nonlinear autoregressive exogenous (NARX) network was proposed in [2] to
predict nitrate concentrations using, water discharge (Q), electrical
conductivity (EC), dissolved oxygen (DO) and temperature (temp). A NARX
network is a kind of neural network which uses values previously predicted by
the network to aid in future predictions, this architecture is typically used
in time-series problems. The Kreutz Creek data-set used daily sampled values
collected from May 2020 to January 2021. This model was trained for 6 months
and was able to achieve overall accurate results for the following 60 days
($R^{2}=0.77,MAE=0.95mg/l,RMSE=1.16mg/l,RAE=48.21\%$), with the best results
within the first 30 days. This paper concluded that to achieve accurate
results at least 2 years of training data is required, in particular if only
water discharge is used.

Generalised Additive Mixed Model (GAMM): Kermorvant et al [8], implemented a
GAMM to explain the nonlinear relationships at each site between nitrate
concentration and conductivity, turbidity, dissolved oxygen, water
temperature, and elevation. The dataset is from the NEON database and
comprises of three locations; the Arikaree River in Colorado, Caribou-Poker
Creeks Research Watershed in Alaska, and Lewis Run in Virginia. Measurements
were taken every 15 minutes of NO3, EC, DO, Water Level and temperature over
two years. There is some missing data, which is largely due to the weather
events e.g. The river at Caribou freezes from approximately October to May
each year and NEON removes most sensors from the site to prevent damage or
loss. This study, emphasises the need to have a model for each river location
i.e. At Caribou, nitrate, dissolved oxygen, specific conductance and turbidity
increased while temperature decreased during the afternoon and the night. At
Arikaree, the diel patterns exhibited comparatively more variation than at
Caribou, and both nitrate and dissolved oxygen fluctuated in the opposite
direction (i.e. decreasing at night and increasing during the day). When a
flow event occurred at Arikaree site (i.e. when the elevation level suddenly
rose), nitrate concentration, oxygen concentration and turbidity increased,
while specific conductance and temperature decreased. Conversely, at Lewis
Run, a sudden increase in water level coincided a decrease in nitrate
concentration. At Caribou, the rise in water level was accompanied by an
increase in turbidity, but the relationships between water level and the other
water-quality covariates were more complex. Furthermore, there were
differences in which water-quality co-variate was most beneficial for
predicting nitrate concentrations - with the most important for Arikaree being
water temperature, the most important for Lewis Run being specific conductance
and time and at Caribou the most important was time and water surface
elevation. The result from this work suggest that a single model cannot be
used across multiple sites due to the unique relationships between nitrates
and water-quality variables. Future research may reveal that models fit to
data from sites with more similar land use, climate conditions and flow
regimes are more transferable.

Limitations: It is important to note that some these implementations use data
sampling schemes which cause the training and testing data-sets to be
intertwined (e.g. the train and test data set points are randomly chosen -
meaning that a test point could be chronologially neighbouring two training
points). This introduces look-ahead bias and only determines how well the
model preforms at interpolating values as opposed to how the model might
preforms on unseen data. Some implementations use K-fold (with cross-
validation), which is reasonable - although the amount of training data (4x
the amount of testing data) used in these examples is significant and there
appears to be large variation across each fold - likely due to seasonal
variation. Other implementations have shown the ability to predict high
frequency nitrates values by using low frequency manual sampling (weekly).
This can be useful to greatly augment the temporal resolution of nutrient
data-sets (i.e. if you have discrete manual nitrate samples and you want to
estimate nitrate levels between samples using high-frequency sensor data).

##  1 Linear vs Non-linear Models

Jingshui et al [12] demonstrated the benefits of using a non-linear model to
predict NO3. When detecting nitrates with a multivariable linear regression
model an R values of 0.67, 0.89 were achieved as opposed to R values of 0.99
and 0.99, when using a RF model.

Non-linear machine learning models present some advantages in comparison to
linear regression. There is no need to find obvious relationships between the
target variable and the surrogates, and they allow capturing phenomena that
have a non-linear behaviour, as usually occurs with environmental processes
[1].

Overall it is unlikely that the a linear regression based model will be able
to achieve desirable accuracy for modelling nitrates, this is due to the
complex non-linear relationship between the predictors variables. There is no
available literature to the best of my knowledge suggesting that a purely
linear model will be suitable for our application.

##  2 Surrogate Sensors

Green et al [5], found that EC was the most important independent variable for
predicting nitrate, likely due to the direct dependence of EC on ionic
strength of a solution. Model predictions also were sensitive to the inclusion
of FDOM which is likely due to FDOM being an effective proxy for dissolved
organic carbon. Nitrate predictions were also sensitive to pH, perhaps because
most stream nitrate at Hubbard Brook is produced by nitrification, which is an
acidifying process.

Harrison et al [7], tested dropping different surrogate variables from the
random forest model. It was found that soil moisture had the biggest affect
followed by hydro-static pressure, then EC. Time of day and pH were found to
be insignificant and were not predictors in the best models. EC was the most
important independent variable, likely due to the direct dependence of EC on
ionic strength of a solution. Model predictions also were sensitive to the
inclusion of FDOM which is likely due to FDOM being an effective proxy for
dissolved organic carbon. Nitrate predictions were also sensitive to pH,
prehaps because most stream nitrate at Hubbard Brook is produced by
nitrification, which is an acidifying process. Turbidity was found to be of
little importance, which is likely due to it being a proxy for suspended
material rather than dissolved solutes.

As mentioned earlier Kermorvant et al [8] found the most significant predictor
variable varied greatly between sites, with EC, temperature, time and water
level each being the best predictor for each site.

Due to the complicated non-linear interaction between the predictor variables,
the choice of variables cannot be considered independent. Generally EC has
found to be the most important predictor variable, however this is not always
the case and it can be observed that it is beneficial to capture a range of
water quality parameters. Additionally the work done by Harrison et al, finds
that there is benefit to incorporating soil moisture and hydro-static pressure
sensors, they also find that lagged and delta values are important. It can
also be observed that there is benefits to incorporating time of day and time
of year as predictor variables, however this depends how much consistent
seasonal and daily variation the site has. Further research would help to
understand how well models can generalize to represent a wide range of
catchments.

##  3 Experiments

### Evaluation Metrics

To evaluate the performance of the various nitrate models, different metrics
are used.

R-Squared (A.K.A. NSE): Is a measure (between -1 and 1) of how well the model
can explain the variance of the data (goodness of fit). Where $\bar{y}$ is the
mean of the ground truth data, $y$ is the ground truth value and $\hat{y}$ is
the model prediction. A R-Squared of 0 can be considered taking the mean value
(i.e. anything close to zero or negative is considered poor).

$R^{2}=1-\frac{\sum_{i=1}^{n}(y_{i}-\hat{y})^{2}}{\sum_{i=1}^{n}({y_{i}}-\bar{y})^{2}}$

Mean Absolute Error: Is a measure of the absolute difference between the
predicted value and the ground truth value.

$MAE=\frac{1}{N}\sum_{i=1}^{N}|y_{i}-\hat{y}_{i}|$

Mean Absolute Percentage Error: Is a measure of the absolute difference
between the predicted value and the ground truth value over the ground truth
value.

$MAPE=\frac{1}{N}\sum_{i=1}^{N}\frac{|y_{i}-\hat{y}_{i}|}{\hat{y}_{i}}$

Mean Squared Error: Is a measure of the difference between the predicted value
and the ground truth value but it penalizes larger errors more heavily.

$MSE=\frac{1}{N}\sum_{i=1}^{N}(y_{i}-\hat{y}_{i})^{2}$

Root Mean Squared Error: Is a measure of the difference between the predicted
value and the ground truth value but it penalizes larger errors more heavily
but it takes the root of this value to put it in a more interpretable form.

$RMSE=\sqrt{\frac{1}{N}\sum_{i=1}^{N}(y_{i}-\hat{y}_{i})^{2}}$

Normalized Root Mean Squared Error: Is a measure of the difference between the
predicted value and the ground truth value but it penalizes larger errors more
heavily but it takes the root of this value to put it in a more interpret-able
form (as a percentage). In our experiments we normalize by the mean.

$nRMSE=\frac{1}{\hat{y}}\sqrt{\frac{1}{N}\sum_{i=1}^{N}(y_{i}-\hat{y}_{i})^{2}}$

### Datasets

To be able to undertake experimental testing we collate data-sets from
previous research. The data-sets include the Arikaree River (from NEON) and
the River Enborne dataset.

The Aikaree River is located in the central plains of North Eastern Colorado.
It has a large watershed size of 2631.8 $km^{2}$ (65,332 acres). The area is
sensitive to invasive plant species. The Arikaree is also impacted by
agricultural runoff and cattle grazing. In the winter the river is a free-
flowing stream and dries into standing pools in summer. The area can be
subject to tornados, flooding, blizzards, and severe winter storms. The
Arikaree River flows through a sand-alluvial basin and its bed and banks are
composed mostly of sand. The grassy shores of Arikaree River are dominated by
grasses. Blue grama grass (Bouteloua gracilis), needle-and-thread grass
(Hesperostipa comata), and big bluestem (Andropogon gerardii) grow strong here
among others. Data is sampled at 15-minute intervals and includes; NO3, EC,
DO, pH, chlorophyll, turbidity, fDOM and temperature. This data-set contains
data from 2018-09-13 till 2019-12-31 (474 days) with 14721 valid samples.

Aikaree (n=14721)
---
| NO3 (mg/l) | Turb (FNU) | Temp (C) | EC (uS/cm) | pH | DO (%)
mean | 0.26 | 38.31 | 11.02 | 547.05 | 7.75 | 53.16
std | 0.15 | 228.08 | 9.3 | 46.93 | 0.32 | 23.41
min | 0.006 | 0.01 | -0.644 | 374.00 | 7.18 | 31.65
max | 1.097 | 3797.0 | 31.52 | 709.75 | 8.41 | 112.92

The River Enborne is a relatively rural river, near Brimpton. It has some
small population centres. The relief of the Enborne catchment is characterised
by gently sloping valleys with a maximum altitude of 296 m above sea level.
The dataset contains hourly measurements of total reactive phosphorus,
nitrate, conductivity, temperature, dissolved oxygen, pH and total
chlorophyll. Throughout the monitoring program, weekly “grab-samples” were
collected at the Brimpton site. In detail analysis of this data is conducted
thoroughly in [6]. This data set has 12723 valid samples from 2009-12-08 till
2011-12-19 (859 days) .

Enborne (n=12723)
---
| NO3 ($mg\/l$) | Turb (NTU) | Temp (C) | EC (uS/cm) | pH | DO (%) | Flow ($m^{3}\/s$)
mean | 4.05 | 8.05 | 10.8 | 486.52 | 7.99 | 90.27 | 0.97
std | 0.74 | 10.27 | 4.4 | 79.63 | 0.224 | 15.58 | 1.32
min | 1.69 | 2.00 | 0.30 | 215.00 | 7.24 | 53.00 | 0.10
max | 6.24 | 215.8 | 19.80 | 769.00 | 8.94 | 170.00 | 12.60

###  3.1 How much training data is required?

Due to each river having a unique complex non-linear relationship between the
water quality parameters and the nitrate concentration; it is important to
understand how much data is required to get a model that suitably fits each
river. Additionally it is important to understand what is the best way to
temporally collect the data (i.e. can the data be collected at a high
frequency in one month or does it need collected through the year).

The Aikaree River data-set has many missing rows which is due to the extreme
weather conditions and sensor reliability. Therefore, for the tests conducted
with this data set we remove the cases in which we cannot get at least 50% of
the training samples for that time period.

Although there is multiple options of which model to use for this experiment,
a logical choice is to use a RF model (with 200 trees). As it is straight
forward to implement, able to accurately represent non-linear relationships
and has support from previous literature about it’s viability. Therefore it is
a sensible model to use as a baseline for some initial testing.

For the following experiments we set a amount of days for the training data
and an amount of days for the test days. We then shift the training and
testing days by 1 (move a day forward in time) and repeat this process until
we have reached the end of the data-set. This allows us to perform the
experiment across the full data-set, thereby improving the reliability of our
results. For each split of the data-set we evaluate our model using the prior
mentioned metrics. For the Enborne dataset we use NO3, EC, pH, do and
turbidity as predictor variables.

Enborne Temporal Analysis (30 days of training)
---
Days After: | 0-30 | 30-60 | 60-180 | 180-360
$R^{2}$ | 0.715 | $-0.272$ | 0.028 | 0.039
MAE (mg/l) | 0.176 | 0.308 | 0.423 | 0.485
MAPE (%) | 4.6 | 7.3 | 10.4 | 12.2
MSE | 0.068 | 0.157 | 0.315 | 0.372
RMSE (mg/l) | 0.243 | 0.384 | 0.556 | 0.607
nRMSE (%) | 0.065 | 0.087 | 0.131 | 0.154

As we can see from the table above, the The $R^{2}$ results are acceptable for
the following 0-30 days. However for the remaining time periods, the results
are unacceptable. The $R^{2}$ scores are close to zero and negative, this
means that the model predictions are worse than a horizontal mean line
(average of the training values - $\bar{y}$). Showing that the model is unable
to explain the variability in the data. This is likely due to the seasonal
variability as-well as extreme weather events that occur. However we can see
that our MAE, MAPE, MSE, RMSE and nRMSE all seem to be within acceptable
ranges. This is interesting and shows the effect of having data that has
little std deviation.

## Results

## Conclusion

Current research proposes solutions which appear promising however, only a few
studies determine how well the solution would work by training with data that
precedes the testing data. The research that does propose using training data
that precedes the testing data is only able to achieve acceptable results for
a short period (less than 60 days). Although if the training data was
collected over a long period of time (2 years +) it might be possible to
achieve better long term results. Other research that has proposed to use
K-Fold cross-validation, leads to the training data set being K-1 times bigger
than the testing set (training set is considerably bigger than the testing
set). This is impractical to implement as we require the training data to be
smaller to ensure our solution is cost-effective to implement.

A few solutions provide an interesting idea of using manual grab samples
(usually weekly) in parallel with a machine learning model, to essentially
augment the data from weekly resolution into hourly resolution. This seems to
be reasonably effective, however it seems better suited as a post-processing
stage as a opposed to a solution that can provide real-time information.

Regarding machine learning model choice, Random Forest regression is a good
starting point (XGBoost potential next step). However, the main bottleneck for
any model is getting appropriate data. Getting suitable data for Rivers is
challenging due to their dynamic nature. Furthermore, models fail to
extrapolate when receiving values outside of their training data - meaning
that it would be difficult for a model to work accurately in extreme cases
(unless appropriate training data was provided).

## References

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  * [2] Di Nunno, F., Race, M., Granata, F.: A nonlinear autoregressive exogenous (narx) model to predict nitrate concentration in rivers. Environmental Science and Pollution Research 29(27), 40623–40642 (2022)
  * [3] Diamantopoulou, M.J., Papamichail, D.M., Antonopoulos, V.Z.: The use of a neural network technique for the prediction of water quality parameters. Operational Research 5(1), 115–125 (2005)
  * [4] Dymond, J., Ausseil, A.G., Parfitt, R., Herzig, A., McDowell, R.: Nitrate and phosphorus leaching in new zealand: a national perspective. New Zealand Journal of Agricultural Research 56(1), 49–59 (2013). https://doi.org/10.1080/00288233.2012.747185, https://doi.org/10.1080/00288233.2012.747185
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  * [6] Halliday, S.J., Skeffington, R.A., Bowes, M.J., Gozzard, E., Newman, J.R., Loewenthal, M., Palmer-Felgate, E.J., Jarvie, H.P., Wade, A.J.: The water quality of the river enborne, uk: Observations from high-frequency monitoring in a rural, lowland river system. Water 6(1), 150–180 (2014)
  * [7] Harrison, J.W., Lucius, M.A., Farrell, J.L., Eichler, L.W., Relyea, R.A.: Prediction of stream nitrogen and phosphorus concentrations from high-frequency sensors using random forests regression. Science of the Total Environment 763, 143005 (2021)
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  * [9] Miller, M.P., Tesoriero, A.J., Capel, P.D., Pellerin, B.A., Hyer, K.E., Burns, D.A.: Quantifying watershed-scale groundwater loading and in-stream fate of nitrate using high-frequency water quality data. Water Resources Research 52(1), 330–347 (2016)
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  * [12] Tran, Y.B., Arias-Rodriguez, L.F., Huang, J.: Predicting high-frequency nutrient dynamics in the danube river with surrogate models using sensors and random forest. Frontiers in Water p. 123 (2022)


# On a multi-integral norm defined by weighted sums of log-concave random
vectors

Nikos Skarmogiannis

###### Abstract

Let $C$ and $K$ be centrally symmetric convex bodies in ${\mathbb{R}}^{n}$. We
show that if $C$ is isotropic then

$\|{\bf
t}\|_{C^{s},K}=\int_{C}\cdots\int_{C}\Big{\|}\sum_{j=1}^{s}t_{j}x_{j}\Big{\|}_{K}\,dx_{1}\cdots
dx_{s}\leqslant c_{1}L_{C}(\log n)^{5}\,\sqrt{n}M(K)\|{\bf t}\|_{2}$

for every $s\geqslant 1$ and ${\bf
t}=(t_{1},\ldots,t_{s})\in{\mathbb{R}}^{s}$, where $L_{C}$ is the isotropic
constant of $C$ and $M(K):=\int_{S^{n-1}}\|\xi\|_{K}d\sigma(\xi)$. This
reduces a question of V. Milman to the problem of estimating from above the
parameter $M(K)$ of an isotropic convex body. The proof is based on an
observation that combines results of Eldan, Lehec and Klartag on the slicing
problem: If $\mu$ is an isotropic log-concave probability measure on
${\mathbb{R}}^{n}$ then, for any centrally symmetric convex body $K$ in
${\mathbb{R}}^{n}$ we have that

$I_{1}(\mu,K):=\int_{{\mathbb{R}}^{n}}\|x\|_{K}\,d\mu(x)\leqslant
c_{2}\sqrt{n}(\log n)^{5}\,M(K).$

We illustrate the use of this inequality with further applications.

##  1 Introduction

Let $K$ be a centrally symmetric convex body in ${\mathbb{R}}^{n}$. For any
$s$-tuple ${\cal C}=(C_{1},\ldots,C_{s})$ of centrally symmetric convex bodies
$C_{j}$ of volume $1$ in ${\mathbb{R}}^{n}$, consider the norm on
${\mathbb{R}}^{s}$, defined by

$\|{\bf t}\|_{{\cal
C},K}=\int_{C_{1}}\cdots\int_{C_{s}}\Big{\|}\sum_{j=1}^{s}t_{j}x_{j}\Big{\|}_{K}\,dx_{s}\cdots
dx_{1}$

where ${\bf t}=(t_{1},\ldots,t_{s})$. If ${\cal C}=(C,\ldots,C)$ then we write
$\|{\bf t}\|_{C^{s},K}$ instead of $\|{\bf t}\|_{{\cal C},K}$. A question
posed by V. Milman is to determine if, in the case $C=K$, one has that
$\|\cdot\|_{K^{s},K}$ is equivalent to the standard Euclidean norm up to a
term which is logarithmic in the dimension, and in particular, if under some
cotype condition on the norm induced by $K$ to ${\mathbb{R}}^{n}$ one has
equivalence between $\|\cdot\|_{K^{s},K}$ and the Euclidean norm.

This question was studied by Bourgain, Meyer, V. Milman and Pajor. For
simplicity let us assume that $\mathrm{vol}_{n}(K)=1$ (this is only a matter
of normalization). It was proved in [5] that

$\|{\bf t}\|_{{\cal C},K}\geqslant
c\sqrt{s}\Big{(}\prod_{j=1}^{s}|t_{j}|\Big{)}^{1/s}$

where $c>0$ is an absolute constant. Later, Gluskin and V. Milman obtained a
better lower bound in [14], in fact working in a more general context: Let
$A_{1},\ldots,A_{s}$ be measurable sets of volume $1$ in ${\mathbb{R}}^{n}$
and $K$ be a star body of volume $1$ in ${\mathbb{R}}^{n}$ with $0\in{\rm
int}(K)$. Then,

(1.1) $\|{\bf t}\|_{{\cal A},K}\geqslant c\,\|{\bf t}\|_{2}$

for all ${\bf t}=(t_{1},\ldots,t_{s})\in{\mathbb{R}}^{s}$. Their argument was
based on the Brascamp-Lieb-Luttinger inequality (see also [3, Chapter 4]).

We are mainly interested in upper bounds for the quantity $\|{\bf
t}\|_{C^{s},K}$. Since $\|{\bf t}\|_{C^{s},K}=\|{\bf t}\|_{(TC)^{s},TK}$ for
any $T\in SL(n)$, we may restrict our attention to the case where $C$ is
isotropic (see Section 2 for the definition and background information). In
fact, we are particularly interested in the case where $C$ is isotropic and
$K=C$, which corresponds to V. Milman’s original question.

For any centered log-concave probability measure $\mu$ on ${\mathbb{R}}^{n}$
and any centrally symmetric convex body $K$ in ${\mathbb{R}}^{n}$, consider
the parameter

(1.2) $I_{1}(\mu,K):=\int_{{\mathbb{R}}^{n}}\|x\|_{K}d\mu(x).$

Assume that $C$ is isotropic. Chasapis, Giannopoulos and the author observed
in [7] that one may write

(1.3) $\|{\bf t}\|_{C^{s},K}=\|{\bf t}\|_{2}L_{C}\,I_{1}(\mu_{{\bf t}},K),$

where $\mu_{{\bf t}}$ is an isotropic, compactly supported log-concave
probability measure depending on ${\bf t}$. One may easily check that if $\mu$
is an isotropic log-concave probability measure on ${\mathbb{R}}^{n}$ and $K$
is a centrally symmetric convex body in ${\mathbb{R}}^{n}$ then

$\int_{O(n)}I_{1}(\mu,U(K))\,d\nu(U)=\int_{{\mathbb{R}}^{n}}\int_{O(n)}\|x\|_{U(K)}d\nu(U)\,d\mu(x)=M(K)\int_{{\mathbb{R}}^{n}}\|x\|_{2}d\mu(x)\approx\sqrt{n}M(K)$

where

$M(K):=\int_{S^{n-1}}\|\xi\|_{K}d\sigma(\xi)$

and $\nu,\sigma$ denote the Haar probability measures on $O(n)$ and $S^{n-1}$
respectively. It follows that

(1.4) $\int_{O(n)}\|{\bf
t}\|_{U(C)^{s},K}\,d\nu(U)\approx(L_{C}\,\sqrt{n}M(K))\,\|{\bf t}\|_{2}.$

Therefore, one might hope to obtain a quantity of the order of
$L_{C}\,\sqrt{n}M(K)\,\|{\bf t}\|_{2}$ as an upper estimate for $\|{\bf
t}\|_{C^{s},K}$. Using an argument which goes back to Bourgain (also,
employing Paouris’ inequality and Talagrand’s comparison theorem) Chasapis,
Giannopoulos and the author proved in [7] that if $C$ is an isotropic
centrally symmetric convex body in $\mathbb{R}^{n}$ and $K$ is a centrally
symmetric convex body in $\mathbb{R}^{n}$ then

$\|{\bf t}\|_{C^{s},K}\leqslant
c\,\Big{(}L_{C}\max\Big{\\{}\sqrt[4]{n},\sqrt{\log(1+s)}\Big{\\}}\Big{)}\,\sqrt{n}M(K)\|{\bf
t}\|_{2}$

for every ${\bf t}=(t_{1},\ldots,t_{s})\in\mathbb{R}^{s}$, where $c>0$ is an
absolute constant. Using the basic identity (1.3) they also obtained an
alternative proof of (1.1).

In this note we provide a poly-logarithmic in $n$, and independent from $s$,
upper bound for $\|{\bf t}\|_{C^{s},K}$.

######  Theorem 1.1.

Let $C$ be an isotropic centrally symmetric convex body in $\mathbb{R}^{n}$
and $K$ be a centrally symmetric convex body in $\mathbb{R}^{n}$. Then,

$\|{\bf t}\|_{C^{s},K}\leqslant c_{1}L_{C}(\log n)^{5}\,\sqrt{n}M(K)\|{\bf
t}\|_{2}$

for every ${\bf t}=(t_{1},\ldots,t_{s})\in\mathbb{R}^{s}$, where $c_{1}>0$ is
an absolute constant.

In the case $C=K$, Theorem 1.1 and (1.1) show that, for any centrally
symmetric convex body $K$ of volume $1$ in ${\mathbb{R}}^{n}$,

$c_{1}\|{\bf t}\|_{2}\leqslant\|{\bf t}\|_{K^{s},K}\leqslant c_{2}(\log
n)^{5}\,\sqrt{n}M(K^{\ast})L_{K^{\ast}}\|{\bf t}\|_{2}$

for every $s\geqslant 1$ and every ${\bf
t}=(t_{1},\ldots,t_{s})\in\mathbb{R}^{s}$, where $c_{i}>0$ are absolute
constants and $K^{\ast}$ is an isotropic linear image of $K$ (note that
$L_{K^{\ast}}=L_{K}$). Thus, we have a reduction of V. Milman’s question to
the problem of estimating the parameter $M(K^{\ast})$ for an isotropic
centrally symmetric convex body $K^{\ast}$ in ${\mathbb{R}}^{n}$. One may hope
that $\sqrt{n}M(K^{\ast})L_{K^{\ast}}\leqslant c_{3}(\log n)^{b}$ for some
absolute constant $b>0$, which would completely settle the problem, modulo the
best possible exponent of $\log n$. However, the latter problem remains open;
see Remark 4.2 for a brief review of what is known.

The source of our improved estimate in Theorem 1.1 is the next result which we
believe is of independent interest.

######  Theorem 1.2.

Let $\mu$ be an isotropic log-concave probability measure on
${\mathbb{R}}^{n}$. For any centrally symmetric convex body $K$ in
${\mathbb{R}}^{n}$ we have that

$I_{1}(\mu,K)\leqslant c_{2}\sqrt{n}(\log n)^{5}\,M(K)$

where $c_{2}>0$ is an absolute constant.

Theorem 1.2 follows from the recent developments on the Kannan-Lovász-
Simonovits conjecture and the slicing problem. We simply combine the recent
estimate of Klartag and Lehec [16] with previous results of Eldan [9] and, in
particular, with a theorem of Eldan and Lehec from [10] which now leads to
this strong and general estimate for $I_{1}(\mu,K)$. We explain the details in
Section 3. In Section 4 we combine Theorem 1.2 with the approach and method
that was introduced in [7] to obtain Theorem 1.1.

In Section 5 we make some observations on the geometry of the unit ball
${\mathcal{B}}_{s}:={\mathcal{B}}_{s}(C^{s},K)$ of the norm
$\|\cdot\|_{C^{s},K}$. By the symmetry of $C$ we easily check that
${\mathcal{B}}_{s}$ is an unconditional $1$-symmetric convex body in
${\mathbb{R}}^{s}$. Therefore, one can determine the volume radius of
${\mathcal{B}}_{s}$ up to a $\sqrt{\log s}$-factor: we have

$\frac{c}{\sqrt{\log s}}M({\mathcal{B}}_{s})\leqslant\frac{1}{{\rm
vrad}({\mathcal{B}}_{s})}\leqslant M({\mathcal{B}}_{s})$

for any $s\geqslant 1$, where $c>0$ is an absolute constant. In the case
$s\geqslant c_{1}n$, where $c_{1}>1$ is an absolute constant, a result of
Bourgain, Meyer, V. Milman and Pajor from [5] is equivalent to the assertion

$M({\mathcal{B}}_{s})\geqslant c_{2}L_{C}\sqrt{n}M(K)$

for some absolute constant $c_{2}>0$. Under the same assumption, that
$s\geqslant c_{1}n$, we obtain an upper bound of the same order, and hence
determine $M({\mathcal{B}}_{s})$.

######  Theorem 1.3.

There exist absolute constants $c_{i}>0$ such that: If $C$ is an isotropic
centrally symmetric convex body in ${\mathbb{R}}^{n}$ then, for any centrally
symmetric convex body $K$ in ${\mathbb{R}}^{n}$ and any $s\geqslant c_{1}n$,

$c_{2}L_{C}\sqrt{n}M(K)\leqslant M({\mathcal{B}}_{s})\leqslant
c_{3}L_{C}\sqrt{n}M(K)$

where ${\mathcal{B}}_{s}$ is the unit ball of the norm $\|\cdot\|_{C^{s},K}$
on ${\mathbb{R}}^{s}$.

Combining Theorem 1.3 with Theorem 1.1 we see that if $s\geqslant c_{1}n$ then
the critical dimension $k({\mathcal{B}}_{s})$ of ${\mathcal{B}}_{s}$ (see [2,
Section 5.3]) is close to $s$; more precisely,

$k({\mathcal{B}}_{s})\geqslant c_{4}s/(\log n)^{10}.$

Then, using a well-known deviation inequality for Lipschitz functions on the
Euclidean sphere of ${\mathbb{R}}^{s}$ we get:

######  Corollary 1.4.

There exist absolute constants $c_{i}>0$ such that: If $C$ is an isotropic
centrally symmetric convex body in ${\mathbb{R}}^{n}$ then for any centrally
symmetric convex body $K$ in ${\mathbb{R}}^{n}$ and any $s\geqslant c_{1}n$,
an $s$-tuple ${\bf t}=(t_{1},\ldots,t_{s})$ with $\|{\bf t}\|_{2}=1$ satisfies

$c_{2}L_{C}\sqrt{n}M(K)\|{\bf t}\|_{2}\leqslant\|{\bf t}\|_{{\cal
B}_{s}}\leqslant c_{3}L_{C}\sqrt{n}M(K)\|{\bf t}\|_{2}$

with probability greater than $1-\exp(-c_{4}s/(\log n)^{10})$.

We believe that Theorem 1.2 is a useful result and in Section 6 we illustrate
its use with some additional applications. In the language of convex bodies,
Theorem 1.2 has the following almost immediate reformulation.

######  Theorem 1.5.

Let $C$ and $K$ be two centrally symmetric convex bodies in
${\mathbb{R}}^{n}$. There exists $T\in SL(n)$ such that

$\frac{1}{\mathrm{vol}_{n}(C)}\int_{C}\|x\|_{TK}dx\leqslant c(\log
n)^{10}\left(\frac{\mathrm{vol}_{n}(C)}{\mathrm{vol}_{n}(K)}\right)^{1/n}$

where $c>0$ is an absolute constant.

The inequality of Theorem 1.5 is a reverse form of the well-known fact (see
[18, Section 2.2]) that for any pair of centrally symmetric convex bodies $K$
and $C$ in ${\mathbb{R}}^{n}$ one has

$\frac{1}{\mathrm{vol}_{n}(C)}\int_{C}\|x\|_{K}dx\geqslant\frac{n}{n+1}\left(\frac{\mathrm{vol}_{n}(C)}{\mathrm{vol}_{n}(K)}\right)^{1/n}.$

It generalizes the $MM^{\ast}$-estimate (see [2, Theorem 6.5.1], more
precisely [2, Corollary 6.5.3]) which corresponds to the case $C=B_{2}^{n}$.

Our second application is an affirmative answer to a question from [12] where
Giannopoulos, Paouris and Vritsiou presented a reduction of the slicing
problem to the study of the parameter

$I_{1}(K,Z_{q}^{\circ}(K))=\int_{K}\|\langle\cdot,x\rangle\|_{L_{q}(K)}dx$

(where $\\{Z_{q}(K)\\}_{q\geqslant 1}$ is the family of $L_{q}$-centroid
bodies of $K$). It was shown in [12] that if (for some $1/2\leqslant s<1$) one
had an upper bound of the form

(1.5) $I_{1}(K,Z_{q}^{\circ}(K))\leqslant c_{1}q^{s}\sqrt{n}L_{K}^{2}$

for every isotropic convex body $K$ in ${\mathbb{R}}^{n}$ and all $1\leqslant
q\leqslant n$, then it would follow that

$L_{n}\leqslant\frac{c_{2}\sqrt[4]{n}\log\\!n}{q^{\frac{1-s}{2}}},$

where $L_{n}:=\max\\{L_{K}:K\;\hbox{is an isotropic convex body
in}\,{\mathbb{R}}^{n}\\}$. However, to the best of our knowledge, it is only
known that (1.5) holds true with $s=1$, which via the reduction of [12]
resulted in an estimate of the form $L_{n}=O(\sqrt[4]{n}(\log n)^{b})$. Using
Theorem 1.2 we confirm in Section 6 that (1.5) holds true with $s=1/2$ up to
some factors which are logarithmic in $n$.

######  Theorem 1.6.

Let $K$ be a convex body of volume $1$ in ${\mathbb{R}}^{n}$ with center of
mass at the origin. Then, for every $1\leqslant q\leqslant n$ we have that

$I_{1}(K,Z_{q}^{\circ}(K))\leqslant c\sqrt{qn}(\log n)^{7}L_{K}^{2}$

where $c>0$ is an absolute constant.

Note that Theorem 1.6 establishes (1.5) with $s=\frac{1}{2}$ (up to a factor
which is logarithmic in $n$) in accordance with the fact that the reduction of
the slicing problem in [12] with $q\approx n$ then leads to a poly-logarithmic
in $n$ upper bound for $L_{n}$. Of course, the order has been now reversed
since in the proof of Theorem 1.6 we use the results of Eldan, Lehec and
Klartag as well as E. Milman’s bounds for the mean width of the
$L_{q}$-centroid bodies $Z_{q}(K)$. However, we believe that this application
illustrates, once again, the strength and possible uses of Theorem 1.2.

##  2 Notation and background information

In this section we introduce notation and terminology that we use throughout
this work, and provide background information on isotropic convex bodies. We
write $\langle\cdot,\cdot\rangle$ for the standard inner product in
${\mathbb{R}}^{n}$ and denote the Euclidean norm by $\|\cdot\|_{2}$. In what
follows, $B_{2}^{n}$ is the Euclidean unit ball, $S^{n-1}$ is the unit sphere,
and $\sigma$ is the rotationally invariant probability measure on $S^{n-1}$.
Lebesgue measure in ${\mathbb{R}}^{n}$ is denoted by ${\rm vol}_{n}$. The
Grassmann manifold $G_{n,k}$ of all $k$-dimensional subspaces of
${\mathbb{R}}^{n}$ is equipped with the Haar probability measure $\nu_{n,k}$.
For every $1\leqslant k\leqslant n-1$ and $E\in G_{n,k}$ we write $P_{E}$ for
the orthogonal projection from $\mathbb{R}^{n}$ onto $E$. The letters
$c,c^{\prime},c_{j},c_{j}^{\prime}$ etc. denote absolute positive constants
whose value may change from line to line. Sometimes we might even relax our
notation: $a\lesssim b$ will then mean “$a\leqslant cb$ for some (suitable)
absolute constant $c>0$”, and $a\approx b$ will stand for “$a\lesssim b\land
a\gtrsim b$”. If $A,B$ are sets, $A\approx B$ will similarly state that
$c_{1}A\subseteq B\subseteq c_{2}A$ for some absolute constants
$c_{1},c_{2}>0$.

A convex body in ${\mathbb{R}}^{n}$ is a compact convex set
$C\subset{\mathbb{R}}^{n}$ with non-empty interior. We say that $C$ is
centrally symmetric if $-C=C$, and that $C$ is centered if its barycenter
$\frac{1}{\mathrm{vol}_{n}(C)}\int_{C}x\,dx$ is at the origin. We say that $C$
is unconditional with respect to the standard orthonormal basis
$\\{e_{1},\ldots,e_{n}\\}$ of ${\mathbb{R}}^{n}$ if $x=(x_{1},\ldots,x_{n})\in
C$ implies that $(\epsilon_{1}x_{1},\ldots,\epsilon_{n}x_{n})\in C$ for any
choice of signs $\epsilon_{i}\in\\{-1,1\\}$. This is equivalent to the fact
that, for any choice of signs $\\{\epsilon_{i}\\}_{i=1}^{n}$ and scalars
$t_{i}$, we have

$\Big{\|}\sum_{i=1}^{n}\epsilon_{i}t_{i}e_{i}\Big{\|}=\Big{\|}\sum_{i=1}^{n}t_{i}e_{i}\Big{\|}.$

A centrally symmetric convex body $C$ in ${\mathbb{R}}^{n}$ is called
$1$-symmetric if the norm induced by $K$ to ${\mathbb{R}}^{n}$ is
unconditional and permutation invariant, i.e.

$\Big{\|}\sum_{i=1}^{n}\epsilon_{i}t_{i}e_{\sigma(i)}\Big{\|}=\Big{\|}\sum_{i=1}^{n}t_{i}e_{i}\Big{\|}$

for any choice of scalars $t_{i}$, any choice of signs
$\\{\epsilon_{i}\\}_{i=1}^{n}$ and any permutation $\sigma$ of
$\\{1,\ldots,n\\}$. For any $1\leqslant p\leqslant\infty$, the unit ball
$B_{p}^{n}$ of $\ell_{p}^{n}=({\mathbb{R}}^{n},\|\cdot\|_{p})$ is a
$1$-symmetric convex body.

The radius $R(C)$ of a convex body $C$ is the smallest $R>0$ such that
$C\subseteq RB_{2}^{n}$. The volume radius of $C$ is the quantity ${\rm
vrad}(C)=\left(\mathrm{vol}_{n}(C)/\mathrm{vol}_{n}(B_{2}^{n})\right)^{1/n}$.
Integration in polar coordinates shows that if the origin is an interior point
of $C$ then the volume radius of $C$ can be expressed as

${\rm
vrad}(C)=\Big{(}\int_{S^{n-1}}\|\xi\|_{C}^{-n}\,d\sigma(\xi)\Big{)}^{1/n}$

where $\|\xi\|_{C}=\inf\\{t>0:\xi\in tC\\}$. We also consider the parameter

$M(C)=\int_{S^{n-1}}\|\xi\|_{C}d\sigma(\xi).$

The support function of $C$ is defined by $h_{C}(y):=\max\bigl{\\{}\langle
x,y\rangle:x\in C\bigr{\\}}$, and the mean width of $C$ is the average

$w(C):=\int_{S^{n-1}}h_{C}(\xi)\,d\sigma(\xi)$

of $h_{C}$ on $S^{n-1}$. For notational convenience we write $\overline{C}$
for the homothetic image of volume $1$ of a convex body
$C\subseteq\mathbb{R}^{n}$, i.e. $\overline{C}:=\mathrm{vol}_{n}(C)^{-1/n}C$.

The polar body $C^{\circ}$ of a centrally symmetric convex body $C$ in
${\mathbb{R}}^{n}$ is defined by

$C^{\circ}:=\bigl{\\{}y\in{\mathbb{R}}^{n}:\langle x,y\rangle\leqslant
1\;\hbox{for all}\;x\in C\bigr{\\}}.$

The Blaschke-Santaló inequality states that ${\rm vol}_{n}(C){\rm
vol}_{n}(C^{\circ})\leqslant{\rm vol}_{n}(B_{2}^{n})^{2}$, with equality if
and only if $C$ is an ellipsoid. The reverse Santaló inequality of Bourgain
and V. Milman asserts that there exists an absolute constant $c>0$ such that,
conversely,

$\Big{(}{\rm vol}_{n}(C){\rm vol}_{n}(C^{\circ})\Big{)}^{1/n}\geqslant
c\,\mathrm{vol}_{n}(B_{2}^{n})^{2/n}\approx 1/n.$

A convex body $C$ in ${\mathbb{R}}^{n}$ is called isotropic if it has volume
$1$, it is centered and its inertia matrix is a multiple of the identity
matrix: there exists a constant $L_{C}>0$ such that

(2.1) $\|\langle\cdot,\xi\rangle\|_{L_{2}(C)}^{2}:=\int_{C}\langle
x,\xi\rangle^{2}dx=L_{C}^{2}$

for all $\xi\in S^{n-1}$. The hyperplane conjecture asks if there exists an
absolute constant $A>0$ such that

(2.2) $L_{n}:=\max\\{L_{C}:C\ \hbox{is isotropic in}\
{\mathbb{R}}^{n}\\}\leqslant A$

for all $n\geqslant 1$. Bourgain proved in [4] that $L_{n}\leqslant
c\sqrt[4]{n}\log\\!n$; later, Klartag [15] improved this bound to
$L_{n}\leqslant c\sqrt[4]{n}$. In a breakthrough work, Chen [8] proved that
for any $\epsilon>0$ there exists $n_{0}(\epsilon)\in{\mathbb{N}}$ such that
$L_{n}\leqslant n^{\epsilon}$ for every $n\geqslant n_{0}(\epsilon)$. Very
recently, Klartag and Lehec [16] showed that the hyperplane conjecture and the
stronger Kannan-Lovász-Simonovits isoperimetric conjecture hold true up to a
factor that is poly-logarithmic in the dimension; more precisely, they
achieved the bound $L_{n}\leqslant c(\log n)^{4}$, where $c>0$ is an absolute
constant. This development will be discussed in Section 3 and, together with
previous work of Eldan and Lehec, is the key for our main result.

A Borel measure $\mu$ on $\mathbb{R}^{n}$ is called $\log$-concave if
$\mu(\lambda A+(1-\lambda)B)\geqslant\mu(A)^{\lambda}\mu(B)^{1-\lambda}$ for
any compact subsets $A$ and $B$ of ${\mathbb{R}}^{n}$ and any
$\lambda\in(0,1)$. A function $f:\mathbb{R}^{n}\rightarrow[0,\infty)$ is
called $\log$-concave if its support $\\{f>0\\}$ is a convex set and the
restriction of $\log{f}$ to it is concave. It is known that if a probability
measure $\mu$ is log-concave and $\mu(H)<1$ for every hyperplane $H$, then
$\mu$ has a log-concave density $f_{{\mu}}$. Note that if $C$ is a convex body
in $\mathbb{R}^{n}$ then the Brunn-Minkowski inequality implies that
$\mathbf{1}_{C}$ is the density of a $\log$-concave measure.

If $\mu$ is a log-concave measure on ${\mathbb{R}}^{n}$ with density
$f_{\mu}$, we define the isotropic constant of $\mu$ by

(2.3)
$L_{\mu}:=\left(\frac{\sup_{x\in{\mathbb{R}}^{n}}f_{\mu}(x)}{\int_{{\mathbb{R}}^{n}}f_{\mu}(x)dx}\right)^{\frac{1}{n}}[\det\textrm{Cov}(\mu)]^{\frac{1}{2n}}$

where $\textrm{Cov}(\mu)$ is the covariance matrix of $\mu$ with entries

(2.4)
$\textrm{Cov}(\mu)_{ij}:=\frac{\int_{{\mathbb{R}}^{n}}x_{i}x_{j}f_{\mu}(x)\,dx}{\int_{{\mathbb{R}}^{n}}f_{\mu}(x)\,dx}-\frac{\int_{{\mathbb{R}}^{n}}x_{i}f_{\mu}(x)\,dx}{\int_{{\mathbb{R}}^{n}}f_{\mu}(x)\,dx}\frac{\int_{{\mathbb{R}}^{n}}x_{j}f_{\mu}(x)\,dx}{\int_{{\mathbb{R}}^{n}}f_{\mu}(x)\,dx}.$

We say that a $\log$-concave probability measure $\mu$ on ${\mathbb{R}}^{n}$
is isotropic if it is centered, i.e. if

(2.5) $\int_{\mathbb{R}^{n}}\langle x,\xi\rangle
d\mu(x)=\int_{\mathbb{R}^{n}}\langle x,\xi\rangle f_{\mu}(x)dx=0$

for all $\xi\in S^{n-1}$, and $\textrm{Cov}(\mu)$ is the identity matrix.

Let $C$ be a centered convex body of volume $1$ in ${\mathbb{R}}^{n}$. For
every $q\geqslant 1$, the $L_{q}$-centroid body $Z_{q}(C)$ of $C$ is the
centrally symmetric convex body with support function

(2.6) $h_{Z_{q}(C)}(y)=\left(\int_{C}|\langle x,y\rangle|^{q}dx\right)^{1/q}.$

From Hölder’s inequality it follows that $Z_{1}(C)\subseteq Z_{p}(C)\subseteq
Z_{q}(C)\subseteq Z_{\infty}(C)$ for all $1\leqslant p\leqslant
q\leqslant\infty$, where $Z_{\infty}(C)={\rm conv}\\{C,-C\\}$. Using Borell’s
lemma, one can check that

(2.7) $Z_{q}(C)\subseteq c_{1}\frac{q}{p}Z_{p}(C)$

for all $1\leqslant p<q$. One can also check that $Z_{q}(C)\supseteq
c_{2}Z_{\infty}(C)$ for all $q\geqslant n$.

We refer to the books [2] and [3] for basic facts from asymptotic convex
geometry. We also refer to [18] and [6] for more information on isotropic
convex bodies and log-concave probability measures.

##  3 The estimate of Eldan and Lehec

######  Definition 3.1.

Let $\mu$ be a centered log-concave probability measure on ${\mathbb{R}}^{n}$.
For any convex body $K$ in ${\mathbb{R}}^{n}$ with $0\in{\rm int}(K)$ we
define

$I_{1}(\mu,K):=\int_{{\mathbb{R}}^{n}}\|x\|_{K}d\mu(x).$

In this section we provide an estimate for $I_{1}(\mu,K)$ in the case where
$\mu$ is isotropic.

###### Proof of Theorem 1.2.

Let $\mu$ be an isotropic log-concave probability measure on
${\mathbb{R}}^{n}$. Eldan and Lehec [10] obtained an upper bound for
$I_{1}(\mu,K)$ which involves the constant

$\tau_{n}^{2}=\sup_{\mu}\,\sup_{\xi\in
S^{n-1}}\,\sum_{i,j=1}^{n}{\mathbb{E}}_{\mu}(x_{i}x_{j}\langle
x,\xi\rangle)^{2}$

where the first supremum is over all isotropic log-concave probability
measures $\mu$ on ${\mathbb{R}}^{n}$.

######  Theorem 3.2 (Eldan-Lehec).

Let $\mu$ be an isotropic log-concave probability measure on
${\mathbb{R}}^{n}$. For any centrally symmetric convex body $K$ in
${\mathbb{R}}^{n}$ we have that

$\int_{{\mathbb{R}}^{n}}\|x\|_{K}\,d\mu(x)\leqslant c_{1}\sqrt{\log
n}\,\tau_{n}\int_{{\mathbb{R}}^{n}}\|x\|_{K}\,d\gamma_{n}(x)$

where $\gamma_{n}$ is the standard Gaussian measure on ${\mathbb{R}}^{n}$ and
$c_{1}>0$ is an absolute constant.

A result of Eldan [9] relates the constant $\tau_{n}$ with the thin-shell
constant

$\sigma_{n}=\sup_{\mu}\sqrt{{\rm Var}_{\mu}(|x|)}$

where the supremum is over all isotropic log-concave probability measures
$\mu$ on ${\mathbb{R}}^{n}$. Eldan proved that

$\tau_{n}^{2}\leqslant c_{2}\sum_{k=1}^{n}\frac{\sigma_{k}^{2}}{k}$

where $c_{2}>0$ is an absolute constant. On the other hand, Klartag and Lehec
[16] have obtained a poly-logarithmic in $n$ upper bound for $\sigma_{n}$:
their main result states that

$\sigma_{n}\leqslant c_{3}(\log n)^{4}$

where $c_{3}>0$ is an absolute constant. Combining these estimates, one gets

$\tau_{n}^{2}\leqslant c_{2}\sum_{k=1}^{n}\frac{\sigma_{k}^{2}}{k}\leqslant
c_{4}(\log n)^{9}.$

Therefore, the estimate of Eldan and Lehec immediately implies that

$I_{1}(\mu,K):=\int_{{\mathbb{R}}^{n}}\|x\|_{K}\,d\mu(x)\leqslant c_{5}(\log
n)^{5}\,\int_{{\mathbb{R}}^{n}}\|x\|_{K}\,d\gamma_{n}(x)$

where $c_{5}>0$ is an absolute constant. Finally, integration in polar
coordinates shows that

$\int_{{\mathbb{R}}^{n}}\|x\|_{K}\,d\gamma_{n}(x)\approx\sqrt{n}\int_{S^{n-1}}\|\xi\|_{K}\,d\sigma(\xi)\approx\sqrt{n}M(K)$

and hence the proof of Theorem 1.2 is complete.∎

##  4 Proof of the upper bound

In this section we provide the proof of Theorem 1.1. It combines the approach
of [7] with Theorem 1.2.

###### Proof of Theorem 1.1.

Let $C$ be an isotropic centrally symmetric convex body in ${\mathbb{R}}^{n}$
and $X_{1},\ldots,X_{s}$ be independent random vectors, uniformly distributed
in $C$. For any ${\bf t}=(t_{1}\ldots,t_{s})\in{\mathbb{R}}^{s}$ we write
$\nu_{{\bf t}}$ for the distribution of the random vector
$t_{1}X_{1}+\cdots+t_{s}X_{s}$. Since $\|{\bf t}\|_{C^{s},K}$ is a norm, we
may always assume that $\|{\bf t}\|_{2}=1$. Note that $\nu_{{\bf t}}$ is an
even log-concave probability measure on ${\mathbb{R}}^{n}$ (this is a
consequence of the Prékopa-Leindler inequality; see [2]). We write $g_{{\bf
t}}$ for the density of $\nu_{{\bf t}}$. Our starting point is the next
observation from [7].

######  Lemma 4.1.

For any ${\bf t}=(t_{1}\ldots,t_{s})\in{\mathbb{R}}^{s}$, we write $\nu_{{\bf
t}}$ for the distribution of the random vector $t_{1}X_{1}+\cdots+t_{s}X_{s}$.
Then,

$\|{\bf t}\|_{C^{s},K}=\int_{{\mathbb{R}}^{n}}\|x\|_{K}d\nu_{{\bf t}}(x).$

It is easily verified that the covariance matrix ${\rm Cov}(\nu_{{\bf t}})$ of
$\nu_{{\bf t}}$ is a multiple of the identity: more precisely,

${\rm Cov}(\nu_{{\bf t}})=L_{C}^{2}\,I_{n}.$

It follows that the function $f_{{\bf t}}(x)=L_{C}^{n}g_{{\bf t}}(L_{C}x)$ is
the density of an isotropic log-concave probability measure $\mu_{{\bf t}}$ on
${\mathbb{R}}^{n}$. Indeed, we have

$\int_{{\mathbb{R}}^{n}}f_{{\bf
t}}(x)x_{i}x_{j}\,dx=L_{C}^{n}\int_{{\mathbb{R}}^{n}}g_{{\bf
t}}(L_{C}x)x_{i}x_{j}\,dx=L_{C}^{-2}\int_{{\mathbb{R}}^{n}}g_{{\bf
t}}(y)y_{i}y_{j}\,dy=\delta_{ij}$

for all $1\leqslant i,j\leqslant n$.

######  Note.

It is proved in [7] that if $\|{\bf t}\|_{2}=1$ then $\|g_{{\bf
t}}\|_{\infty}\leqslant e^{n}$. From this inequality we see that

$L_{\mu_{{\bf t}}}=\|f_{{\bf t}}\|_{\infty}^{\frac{1}{n}}=L_{C}\|g_{{\bf
t}}\|_{\infty}^{\frac{1}{n}}\leqslant eL_{C}$

for all ${\bf t}\in{\mathbb{R}}^{s}$ with $\|{\bf t}\|_{2}=1$.

We compute

$\|{\bf t}\|_{C^{s},K}=\int_{{\mathbb{R}}^{n}}\|x\|_{K}\,d\nu_{{\bf
t}}(x)=L_{C}^{-n}\int_{{\mathbb{R}}^{n}}\|x\|_{K}f_{{\bf
t}}(x/L_{C})\,dx=L_{C}\int_{{\mathbb{R}}^{n}}\|y\|_{K}d\mu_{{\bf t}}(y)$

and hence we get

(4.1) $\|{\bf t}\|_{C^{s},K}=L_{C}\,I_{1}(\mu_{{\bf t}},K)$

for all ${\bf t}\in{\mathbb{R}}^{s}$ with $\|{\bf t}\|_{2}=1$. Now, we use
Theorem 1.2 to estimate $I_{1}(\mu_{{\bf t}},K)$. As a result, we obtain the
upper bound

$\|{\bf t}\|_{C^{s},K}\leqslant c_{1}L_{C}(\log n)^{5}\,\sqrt{n}M(K)$

which is the assertion of Theorem 1.1.∎

######  Remark 4.2.

We mentioned in the introduction that the question to estimate the parameter
$M(K)$ for an isotropic centrally symmetric convex body $K$ in
${\mathbb{R}}^{n}$ remains open. The currently best known estimates are due to
Giannopoulos and E. Milman (see [11]; also, [13] for previous work on this
question). In view of the recent developments on the slicing problem, we
briefly recall the main computations in their argument and make some slight
modifications to get the next result.

######  Theorem 4.3 (Giannopoulos-E. Milman).

Let $K$ be an isotropic centrally symmetric convex body in ${\mathbb{R}}^{n}$.
Then,

(4.2) $M(K)\leqslant\frac{cn^{1/3}(\log n)^{5/3}}{\sqrt{n}L_{K}}$

where $c>0$ is an absolute constant.

###### Proof.

It is proved in [11, Corollary 4.4] that for every centrally symmetric convex
body $K$ in ${\mathbb{R}}^{n}$ with $K\supseteq rB_{2}^{n}$, one has

(4.3) $\sqrt{n}M(K)\leqslant
c_{1}\sum_{k=1}^{n}\frac{1}{\sqrt{k}}\min\left\\{\frac{1}{r},\frac{n}{k}\log\Big{(}e+\frac{n}{k}\Big{)}\frac{1}{v_{k}^{-}(K)}\right\\}$

where

$v^{-}_{k}(K):=\inf\left\\{{\rm vrad}(P_{E}(K)):E\in G_{n,k}\right\\}.$

Now, since $K$ is isotropic we know that $K\supseteq L_{K}B_{2}^{n}$,
therefore we may use (4.3) with $r=L_{K}$. We also know (see [6, Prooposition
5.1.15]) that

$\mathrm{vol}(K\cap E^{\perp})^{1/k}\leqslant c_{2}\frac{L_{k}}{L_{K}}$

for every $E\in G_{n,k}$. Applying the Rogers-Shephard inequality we see that
for every $E\in G_{n,k}$ we have

$\mathrm{vol}(P_{E}(K))^{1/k}\geqslant\frac{1}{\mathrm{vol}(K\cap
E^{\perp})^{1/k}}\geqslant c_{3}\frac{L_{K}}{L_{k}}$

for every $E\in G_{n,k}$. Therefore, ${\rm vrad}(P_{E}(K))\geqslant
c_{4}\sqrt{k}L_{K}/L_{k}$, which gives

$v_{k}^{-}(K)\geqslant c_{4}\sqrt{k}L_{K}/L_{k}.$

Set $k_{n}=n^{2/3}(\log n)^{10/3}$. Inserting the above estimates into (4.3)
and using the fact that $L_{k}=O((\log n)^{4})$, we get

$\displaystyle\sqrt{n}M(K)$
$\displaystyle\leqslant\frac{c_{5}}{L_{K}}\sum_{k=1}^{n}\frac{1}{\sqrt{k}}\min\left\\{1,\frac{n(\log
n)^{5}}{k^{3/2}}\right\\}\leqslant\frac{c_{6}}{L_{K}}\left(\sum_{k=1}^{k_{n}}\frac{1}{\sqrt{k}}+\sum_{k=k_{n}}^{n}\frac{n(\log
n)^{5}}{k^{2}}\right)$ $\displaystyle\approx\frac{n^{1/3}(\log
n)^{5/3}}{L_{K}}.$

This proves the theorem.∎

##  5 Geometry of the unit ball

Let $C$ and $K$ be centrally symmetric convex bodies in ${\mathbb{R}}^{n}$ and
assume that $C$ is isotropic. In this section we fix $C$ and $K$, and write
${\mathcal{B}}_{s}:={\mathcal{B}}_{s}(C^{s},K)\subset{\mathbb{R}}^{s}$ for the
unit ball of the norm

$\|{\bf
t}\|_{C^{s},K}=\int_{C}\cdots\int_{C}\Big{\|}\sum_{j=1}^{s}t_{j}x_{j}\Big{\|}_{K}\,dx_{s}\cdots
dx_{1}.$

By the symmetry of $C$ we easily check that $\|\cdot\|_{C^{s},K}$ is an
unconditional norm. Moreover, it is $1$-symmetric. Therefore,
${\mathcal{B}}_{s}$ is a $1$-symmetric convex body in ${\mathbb{R}}^{s}$ and
hence it is known that it satisfies the $MM^{\ast}$-estimate

(5.1) $M({\mathcal{B}}_{s})w({\mathcal{B}}_{s})\leqslant c\sqrt{\log s}$

where $c>0$ is an absolute constant (see [19]). We also recall the general
bounds

(5.2) $\frac{1}{M({\mathcal{B}}_{s})}\leqslant{\rm
vrad}({\mathcal{B}}_{s})\leqslant w({\mathcal{B}}_{s})$

that hold for any convex body which contains the origin as an interior point.

######  Lemma 5.1.

For any $s\geqslant 1$ we have that

$\frac{c_{1}}{\sqrt{\log
s}}M({\mathcal{B}}_{s})\leqslant\frac{c_{2}}{w({\mathcal{B}}_{s})}\leqslant\frac{1}{\sqrt{s}}\Big{\|}\sum_{i=1}^{s}e_{i}\Big{\|}_{{\mathcal{B}}_{s}}\approx\frac{1}{{\rm
vrad}({\mathcal{B}}_{s})}\leqslant M({\mathcal{B}}_{s})$

where $c_{1},c_{2}>0$ are absolute constants.

###### Proof.

Since ${\mathcal{B}}_{s}$ is $1$-symmetric, we know that
$\overline{{\mathcal{B}}_{s}}:=|{\mathcal{B}}_{s}|^{-1/s}{\mathcal{B}}_{s}$ is
isotropic. A well-known result of Bobkov and Nazarov (see [6, Chapter 4])
shows that

$c_{1}|{\mathcal{B}}_{s}|^{1/s}B_{\infty}^{s}\subseteq{\mathcal{B}}_{s}\subseteq
c_{2}s|{\mathcal{B}}_{s}|^{1/s}B_{1}^{s}.$

Equivalently, for all $t_{1},\ldots,t_{s}\in{\mathbb{R}}$ we have that

$c_{2}^{\prime}|{\mathcal{B}}_{s}|^{-1/s}\cdot\frac{1}{s}\sum_{i=1}^{s}|t_{i}|\leqslant\Big{\|}\sum_{i=1}^{s}t_{i}e_{i}\Big{\|}_{{\mathcal{B}}_{s}}\leqslant
c_{1}^{\prime}|{\mathcal{B}}_{s}|^{-1/s}\cdot\max_{1\leqslant i\leqslant
s}|t_{i}|.$

Choosing $t_{1}=\cdots=t_{s}=1$ we get the equivalence

$\frac{1}{\sqrt{s}}\Big{\|}\sum_{i=1}^{s}e_{i}\Big{\|}_{{\mathcal{B}}_{s}}\approx\frac{1}{{\rm
vrad}({\mathcal{B}}_{s})}.$

The remaining assertions of the lemma follow from (5.1) and (5.2). ∎

The following theorem is proved in [5].

######  Theorem 5.2 (Bourgain-Meyer-V. Milman-Pajor).

There exist absolute constants $c_{i}>0$ such that, if $C$ is an isotropic
centrally symmetric convex body in ${\mathbb{R}}^{n}$ then for any centrally
symmetric convex body $K$ in ${\mathbb{R}}^{n}$ and any $s\geqslant c_{1}n$,

$\int_{C}\cdots\int_{C}\int_{\Omega}\Big{\|}\sum_{j=1}^{s}g_{j}(\omega)x_{j}\Big{\|}_{K}\,d\omega\,dx_{s}\cdots
dx_{1}\geqslant c_{2}\sqrt{s}L_{C}\sqrt{n}M(K).$

######  Note.

We easily check that

$\displaystyle\int_{C}\cdots\int_{C}\int_{\Omega}\Big{\|}\sum_{j=1}^{s}g_{j}(\omega)x_{j}\Big{\|}_{K}\,d\omega\,dx_{s}\cdots
dx_{1}$
$\displaystyle=\int_{\Omega}\Big{\|}\sum_{j=1}^{s}g_{j}(\omega)e_{j}\Big{\|}_{C^{s},K}d\omega$
$\displaystyle=\int_{{\mathbb{R}}^{s}}\|y\|_{C^{s},K}d\gamma_{s}(y)\approx\sqrt{s}M({\mathcal{B}}_{s}).$

Therefore, Theorem 5.2 states that if $s\geqslant c_{1}n$ then

(5.3) $M({\mathcal{B}}_{s})\geqslant c_{2}L_{C}\sqrt{n}M(K)$

for some absolute constant $c_{2}>0$. We will show that (for the same values
of $s$) a reverse inequality is also true.

######  Theorem 5.3.

There exist absolute constants $c_{i}>0$ such that: If $C$ is an isotropic
centrally symmetric convex body in ${\mathbb{R}}^{n}$ then for any centrally
symmetric convex body $K$ in ${\mathbb{R}}^{n}$ and any $s\geqslant c_{1}n$,

$c_{2}L_{C}\sqrt{n}M(K)\leqslant M({\mathcal{B}}_{s})\leqslant
c_{3}L_{C}\sqrt{n}M(K),$

where ${\mathcal{B}}_{s}$ is the unit ball of the norm $\|\cdot\|_{C^{s},K}$
on ${\mathbb{R}}^{s}$.

###### Proof.

We shall use a well-known result of Adamczak, Litvak, Pajor and Tomczak-
Jaegermann from [1]: For every $\epsilon>0$ and $t\geqslant 1$ there exists a
constant $C=C(t,\epsilon)>0$ with the following property: if $s\geqslant
C(t,\epsilon)n$ and $X_{1},\ldots,X_{s}$ are independent random vectors in
${\mathbb{R}}^{n}$ with the same isotropic log-concave distribution, then we
have

$\Big{\|}\frac{1}{s}\sum_{j=1}^{s}X_{j}\otimes
X_{j}-I_{n}\Big{\|}\leqslant\epsilon$

with probability greater than $1-\exp(-ct\sqrt{n})$. Choosing
$\epsilon=\frac{1}{2}$, $t=1$ and applying this result we get that there
exists an absolute constant $c_{1}>0$ such that if $s\geqslant c_{1}n$ then
there exists ${\cal A}\subset C^{s}$ of measure $|{\cal
A}|>1-e^{-c_{2}\sqrt{n}}$ such that for any $s$-tuple
$(x_{1},\ldots,x_{s})\in{\cal A}$ we have that

(5.4) $\frac{1}{2}sL_{C}^{2}\|z\|_{2}^{2}\leqslant\sum_{j=1}^{s}\langle
x_{j},z\rangle^{2}\leqslant\frac{3}{2}sL_{C}^{2}\|z\|_{2}^{2}$

for every $z\in{\mathbb{R}}^{n}$. Fix $(x_{1},\ldots,x_{s})\in{\cal A}$. We
have

$\frac{1}{2}sL_{C}^{2}\sum_{j=1}^{n}\langle
z,e_{j}\rangle^{2}\leqslant\sum_{j=1}^{s}\langle
z,x_{j}\rangle^{2}\leqslant\frac{3}{2}sL_{C}^{2}\sum_{j=1}^{n}\langle
z,e_{j}\rangle^{2}$

for all $z\in{\mathbb{R}}^{n}$. Applying Slepian’s comparison principle (see
[20] or [2, Chapter 9]) for the Gaussian processes

$X_{z}:=\Big{\langle}\sum_{j=1}^{s}g_{j}x_{j},z\Big{\rangle}\quad\hbox{and}\quad
Y_{z}:=\sum_{j=1}^{s}z_{j}g_{j}^{\prime}\qquad(z\in K^{\circ})$

where $g_{j},g_{j}^{\prime}$ are independent standard Gaussian random
variables, we see that

(5.5)
$\int_{\Omega}\Big{\|}\sum_{j=1}^{s}g_{j}(\omega)x_{j}\Big{\|}_{K}\,d\omega\leqslant
c_{3}\sqrt{s}L_{C}\int_{\Omega}\Big{\|}\sum_{j=1}^{n}g_{j}^{\prime}(\omega)e_{j}\Big{\|}_{K}\,d\omega\leqslant
c_{4}\sqrt{s}L_{C}\sqrt{n}M(K)$

for every $(x_{1},\ldots,x_{s})\in{\cal A}$. On the other hand, if
$(x_{1},\ldots,x_{s})\notin{\cal A}$ then we may write

$\sum_{j=1}^{s}\langle
x_{j},z\rangle^{2}\leqslant\sum_{j=1}^{s}\|x_{j}\|_{2}^{2}\|z\|_{2}^{2}\leqslant
sR(C)^{2}\|z\|_{2}^{2}\leqslant c_{5}sn^{2}L_{C}^{2}\|z\|_{2}^{2},$

recalling that $R(C)\leqslant c_{6}nL_{C}$ because $C$ is isotropic. Working
as before, we obtain the bound

(5.6)
$\int_{\Omega}\Big{\|}\sum_{j=1}^{s}g_{j}(\omega)x_{j}\Big{\|}_{K}\,d\omega\leqslant
c_{7}\sqrt{s}nL_{C}\sqrt{n}M(K)$

for every $(x_{1},\ldots,x_{s})\notin{\cal A}$. Integrating on ${\cal A}$ and
using (5.5) and (5.6) we finally get

$\displaystyle\sqrt{s}M({\mathcal{B}}_{s})$
$\displaystyle\approx\int_{C}\cdots\int_{C}\int_{\Omega}\Big{\|}\sum_{j=1}^{s}g_{j}(\omega)x_{j}\Big{\|}_{K}\,d\omega\,dx_{s}\cdots
dx_{1}$ $\displaystyle=\int_{{\cal
A}}\int_{\Omega}\Big{\|}\sum_{j=1}^{s}g_{j}(\omega)x_{j}\Big{\|}_{K}\,d\omega+\int_{C^{s}\setminus{\cal
A}}\int_{\Omega}\Big{\|}\sum_{j=1}^{s}g_{j}(\omega)x_{j}\Big{\|}_{K}\,d\omega$
$\displaystyle\leqslant
c_{8}\sqrt{s}\big{(}1+n\cdot\exp(-c\sqrt{n})\big{)}L_{C}\sqrt{n}M(K)\leqslant
c_{9}\sqrt{s}L_{C}\sqrt{n}M(K),$

which shows that

$M({\mathcal{B}}_{s})\leqslant c_{10}L_{C}\sqrt{n}M(K).$

The proof of (5.3), which was already obtained in [5], follows by a similar
(and simpler) argument if we start from the left-hand side inequality of (5.4)
.∎

######  Remark 5.4.

Combining Theorem 5.3 with Theorem 1.1 we obtain further information in the
case $s\geqslant c_{1}n$. Note that the critical dimension of
${\mathcal{B}}_{s}$ (see [2, Section 5.3]) satisfies
$k({\mathcal{B}}_{s})\approx s\big{(}M/b\big{)}^{2}$ where
$M=M({\mathcal{B}}_{s})\gtrsim L_{C}\sqrt{n}M(K)$ by (5.3) and $b$ is the best
constant for which $\|{\bf t}\|_{C^{s},K}\leqslant b\|{\bf t}\|_{2}$ for all
${\bf t}\in{\mathbb{R}}^{s}$. From Theorem 1.1 we know that $b\lesssim
c_{1}L_{C}(\log n)^{5}\,\sqrt{n}M(K)$, and hence

$k({\mathcal{B}}_{s})\geqslant cs/(\log n)^{10}.$

A standard application of the deviation inequality for Lipschitz functions on
the Euclidean sphere of ${\mathbb{R}}^{s}$ (see [2, Proposition 5.2.4]) shows
that an $s$-tuple ${\bf t}=(t_{1},\ldots,t_{s})$ with $\|{\bf t}\|_{2}=1$
satisfies

$\frac{1}{2}M({\cal B}_{s})\|{\bf t}\|_{2}\leqslant\|{\bf t}\|_{{\cal
B}_{s}}\leqslant\frac{3}{2}M({\cal B}_{s})\|{\bf t}\|_{2}$

with probability greater than $1-\exp(-c_{3}s/(\log n)^{10})$ and hence, by
Theorem 5.3,

$\|{\bf t}\|_{{\cal B}_{s}}\approx L_{C}\sqrt{n}M(K)\|{\bf t}\|_{2}$

with probability greater than $1-\exp(-c_{3}s/(\log n)^{10})$ on the sphere of
${\mathbb{R}}^{s}$.

##  6 Further remarks and applications

We believe that Theorem 1.2 is a useful result that should find several
applications. As a first example, we prove Theorem 1.5: If $C$ and $K$ are two
centrally symmetric convex bodies in ${\mathbb{R}}^{n}$ then there exists
$T\in SL(n)$ such that

$\frac{1}{\mathrm{vol}_{n}(C)}\int_{C}\|x\|_{TK}dx\leqslant c(\log
n)^{9}\left(\frac{\mathrm{vol}_{n}(C)}{\mathrm{vol}_{n}(K)}\right)^{1/n}$

where $c>0$ is an absolute constant.

###### Proof of Theorem 1.5.

By homogeneity we may assume that $\mathrm{vol}_{n}(C)=1$. We may find
$T_{1}\in SL(n)$ such that $C_{1}:=T_{1}(C)$ is isotropic. Consider the
isotropic log-concave probability measure $\nu_{C}$ with density
$L_{C}^{n}{\mathbf{1}}_{K/L_{C}}$. Direct computation shows that

$\int_{C_{1}}\|x\|_{T_{1}T(K)}dx=I_{1}(\nu_{C_{1}},T_{1}T(K))L_{C}$

for every $T\in SL(n)$. Using Theorem 1.2 we get

$\int_{C}\|x\|_{TK}dx=\int_{C_{1}}\|x\|_{T_{1}T(K)}dx\leqslant
c_{1}\sqrt{n}(\log n)^{5}L_{C}\,M(T_{1}T(K))$

for every $T\in SL(n)$, where $c_{1}>0$ is an absolute constant. From Pisier’s
inequality (see [2, Corollary 6.5.3]) and the Blaschke-Santaló inequality we
may choose $T$ so that

$\displaystyle M(T_{1}T(K))$
$\displaystyle=w((T_{1}T(K))^{\circ})=w((T_{1}T)^{-\ast}(K^{\circ}))\leqslant
c_{2}\sqrt{n}(\log n)\mathrm{vol}_{n}(K^{\circ})^{1/n}$
$\displaystyle\leqslant c_{3}\sqrt{n}(\log
n)\frac{\mathrm{vol}_{n}(B_{2}^{n})^{2/n}}{\mathrm{vol}_{n}(K)^{1/n}}\approx\frac{\log
n}{\sqrt{n}}\mathrm{vol}_{n}(K)^{-1/n},$

since $\mathrm{vol}_{n}(B_{2}^{n})^{2/n}\approx 1/n$. Combining the above we
see that there exists $T\in SL(n)$ such that

$\int_{C}\|x\|_{TK}dx\leqslant c(\log n)^{6}L_{C}\,\mathrm{vol}_{n}(K)^{-1/n}$

and the result follows from the bound $L_{C}\leqslant L_{n}=O((\log n)^{4})$.∎

As a second application we prove Theorem 1.6 which gives an answer to a
question from [12] regarding the parameter

$I_{1}(K,Z_{q}^{\circ}(K)):=\int_{K}\|\langle\cdot,x\rangle\|_{L_{q}(K)}dx.$

###### Proof of Theorem 1.6.

Note that $I_{1}(K,Z_{q}^{\circ}(K))$ is invariant under invertible linear
transformations of $K$ and hence we may assume that $K$ is isotropic. As in
the proof of Theorem 1.5, consider the isotropic log-concave probability
measure $\nu_{K}$ with density $L_{K}^{n}{\mathbf{1}}_{K/L_{K}}$. Direct
computation shows that

$I_{1}(K,Z_{q}^{\circ}(K))=I_{1}(\nu_{K},Z_{q}^{\circ}(K))L_{K}.$

Using Theorem 1.2 we immediately see that

$I_{1}(\nu_{K},Z_{q}^{\circ}(K))\leqslant c_{1}(\log
n)^{5}\,\int_{{\mathbb{R}}^{n}}\|x\|_{Z_{q}(K)^{\circ}}\,d\gamma_{n}(x)\leqslant
c_{2}\sqrt{n}(\log n)^{5}w(Z_{q}(K)).$

We use E. Milman’s estimates [17] on the mean width $w(Z_{q}(K))$ of the
$L_{q}$-centroid bodies $Z_{q}(K)$ of an isotropic convex body $K$ in
${\mathbb{R}}^{n}$.

######  Theorem 6.1 (E. Milman).

Let $K$ be an isotropic convex body in ${\mathbb{R}}^{n}$. Then, for all
$1\leqslant q\leqslant n$ one has

(6.1) $w(Z_{q}(K))\leqslant
c_{1}\sqrt{q}\log(1+q)\max\left\\{\frac{\sqrt{q}\log(1+q)}{\sqrt{n}},1\right\\}L_{K}$

where $c_{1}>0$ is an absolute constant.

Combining the above we get

$I_{1}(K,Z_{q}^{\circ}(K))=I_{1}(\nu_{K},Z_{q}^{\circ}(K))L_{K}\leqslant
c_{1}\sqrt{qn}(\log
n)^{5}\log(1+q)\max\left\\{\frac{\sqrt{q}\log(1+q)}{\sqrt{n}},1\right\\}L_{K}^{2}$

for all $1\leqslant q\leqslant n$ and the theorem follows.∎

Acknowledgement. The author acknowledges support by the Hellenic Foundation
for Research and Innovation (H.F.R.I.) under the “First Call for H.F.R.I.
Research Projects to support Faculty members and Researchers and the
procurement of high-cost research equipment grant” (Project Number: 1849).

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Keywords: convex bodies, log-concave probability measures, weighted sums of
random vectors, isotropic position.
2010 MSC: Primary 46B06; Secondary 52A23, 52A40, 60D05.

Nikos Skarmogiannis: Department of Mathematics, National and Kapodistrian
University of Athens, Panepistimioupolis 157-84, Athens, Greece.

E-mail<EMAIL_ADDRESS>