# Implicit Assimilation of Sparse In Situ Data
for Dense & Global Storm Surge Forecasting

Patrick Ebel∗
<EMAIL_ADDRESS>Brandon Victor†
<EMAIL_ADDRESS>Peter Naylor∗
<EMAIL_ADDRESS>Gabriele Meoni∗,∘
<EMAIL_ADDRESS>Federico Serva‡
<EMAIL_ADDRESS>Rochelle Schneider∗
<EMAIL_ADDRESS>∗European Space Agency, $\Phi$-lab † La Trobe
University ∘European Space Agency, ACT ‡ Consiglio Nazionale delle Ricerche

###### Abstract

Hurricanes and coastal floods are among the most disastrous natural hazards.
Both are intimately related to storm surges, as their causes and effects,
respectively. However, the short-term forecasting of storm surges has proven
challenging, especially when targeting previously unseen locations or sites
without tidal gauges. Furthermore, recent work improved short and medium-term
weather forecasting but the handling of raw unassimilated data remains non-
trivial. In this paper, we tackle both challenges and demonstrate that neural
networks can implicitly assimilate sparse in situ tide gauge data with coarse
ocean state reanalysis in order to forecast storm surges. We curate a global
dataset to learn and validate the dense prediction of storm surges, building
on preceding efforts. Other than prior work limited to known gauges, our
approach extends to ungauged sites, paving the way for global storm surge
forecasting.

##  1 Introduction

Spatio-temporal deep neural network Input sequence of length $T$Output for
lead time $L$$(T+L)$ Sparse in-situ tide gauges Coarse ocean state reanalysis
Atmospheric state reanalysis Densified storm surge forecast Auxiliary coarse
state forecast $1$$2$$...$$T$ weight sharing temporal conditioning lead time
$L$ weight sharing

Figure 1: Overview: Our approach provides densified high-resolution storm
surge forecasts (top) by implicitly assimilating inputs of sparse in situ tide
gauge time series (top) with paired sequences of ocean (center) and weather
state (bottom) re-analysis products. For additional supervision, coarse ocean
state reanalysis maps (bottom) at coinciding lead time are also predicted.

Space-borne Earth observation allows for large-scale monitoring of our planet,
its atmosphere and events such as natural hazards that may pose significant
threat to human life. While the strengths of satellite imagery are its broad
spatial extent, its spatio-temporal resolution is inferior compared to on-site
measurements. In contrast, in situ sensors may provide (sub-)hourly recordings
at highest accuracy, yet they are sparsely deployed and thus lack spatial
coverage. Fusing both kinds of data at a global scale holds promises, but
harmonizing the sensors in a manner suitable for neural networks to process is
an open research direction. A well-established paradigm tackling this issue in
the context of weather analysis is that of data assimilation [13, 10].
However, it is computationally costly and not easily approachable. In this
work, we address the challenge of global storm surge forecasting by implicitly
assimilating sparse and raw tide gauge data with coarse weather and ocean
state reanalysis products. Storm surges are extreme weather-driven ocean
dynamics superimposed on the mean sea level and tidal rhythms, which can cause
coastal floods. Scientific consensus is that the coming decades will bring a
sharp increase of coastal hazards due to climate change-caused rise in mean
sea levels [17, 22, 36], aggravated by land subsidence [39] and more intense
extreme storm events [8, 2]. Our work on storm surge forecasting is motivated
to address such hazards, aligning with the United Nation Sustainable
Development Goals 11.5 & 13 [31, 28]. Particularly, our approach is inspired
by recent advances in weather forecasting that allow generalization to
previously unencountered or ungauged sites, which may especially benefit
under-served communities with less access to well-maintained in situ
measurement infrastructure. To empower worldwide AI-driven surge forecasting,
we curate a novel global and multiple decades spanning dataset of in situ
tidal gauge records, paired with weather and ocean state reanalysis, all
preprocessed according to the best domain-specific practices. We highlight the
dataset’s worth by benchmarking a diverse landscape of approaches including
conventional forecasting techniques, an operational numerical model, state-of-
the-art deep neural networks, a recent vision transformer for weather
forecasting and our enhanced adaptation of a popular lightweight temporal
attention network. While we evince the competitiveness of the latter model,
the main objective of our experiments is to demonstrate the predictability of
storm surges at previously unencountered or altogether ungauged locations, an
aim whose feasibility has been questioned in prior work [4, 37].

In sum, our main contributions are two-fold:

  * •

We introduce a novel, global and multi-decadal dataset of in situ ocean surge
time series, paired with atmospheric and ocean state reanalysis products,
spuring and facilitating further research on this critical matter.

  * •

We demonstrate that precise and dense storm surge forecasts can be obtained by
fusing sparse in situ data of coastal tide gauges with coarse atmosphere and
ocean reanalysis. Critically, our forecasts extend to previously unseen gauges
and entirely ungauged locations, which may benefit under-served communities.

Figure 2:  Data. Green & orange dots denote storm surge time series locations
with records in 1979-2019, as pre-processed from the GESLA-3 collection of
tide gauges [9]. Dark lines indicate hurricane tracks in 2014-2019 as indexed
by IBTRaCS [18]. Pink markers highlight test split gauges, biased to points of
landfall. Visualizations of the ERA5 grid and the irregular GTSM grid are
omitted for brevity.

##  2 Related Work

###  2.1 Short-to-medium Range Weather Forecasting

Recent work marked notable progress on numerical weather prediction [34, 26,
3, 20]. Such models typically rely on atmospheric initial values from a
reanalysis product such as ERA5 [10], i.e. best estimates derived by updating
prior knowledge with multi-source weather observations. Contrarily, the
contribution of [1] proposes an implicit assimilation approach to fuse in situ
weather radar station data with coarse resolution reanalysis products,
yielding dense and skillful rainfall forecasts over the United States. While
we draw inspiration from this approach, our focus is on the global coastlines
to model marine dynamics. Technically, our approach deviates from the
aforementioned ones by processing local patches of data instead of a coarsely
resolved global context in a single forward pass of the network, and is thus
significantly more lightweight.

###  2.2 Storm Surge Forecasting

Operational storm surge forecasting pre-dates deep learning, with early
techniques explicitly modeling the physics of maritime dynamics with a focus
on particular ocean basins [21, 35]. Of particular interest is the Global Tide
and Surge Model (GTSM), a hydrodynamic model forced with ERA5 to globally
predict surge on an irregular grid. In this study, we built upon coarsely
resolved GTSM ocean state analysis [23] to drive the assimilation of raw in
situ data. That is, we fuse the reanalysis product with accurate but sparse in
situ records for improved and densified surge predictions.

Initial efforts for global storm surge modeling via deep learning are given by
[4, 37]. Both studies are limited to temporal generalization, i.e. they
evaluate on gauges trained upon and solely generalize to future time points of
these gauges. In contrast, our data and approach enable to generalize in both
time and space: Relatedly, recent work [25] proved the feasibility of river
streamflow predictions at ungauged basins, in the spirit of which we
generalize coastal storm surge forecasting to unseen shores. This defeats the
prevailing wisdom that ocean modeling necessitates at least 6-7 prior years of
training data at any site of interest [4, 37].

##  3 Data

We collect a new global multi-decadal dataset combining co-registered
atmosphere reanalysis, ocean state reanalysis and pre-processed in situ tide
gauge measurements. All data is sampled to an hourly frequency with dates
ranging from the beginning of $1979$ to the start of $2019$, and gridded at
$0.025^{\circ}$ spatial resolution. The atmosphere reanalysis denotes best
estimates of historical mean sea level pressure as well as 10 metre U and V
wind components (’msl’, ’u10’ and ’v10’, respectively) at about $30$ km
resolution, as provided by the ERA5 catalogue [10]. The ocean-state model
forced by ERA5 meteorology provides the storm surge residual based upon the
irregularly gridded Deltares Global Tide and Surge Model (GTSM) forced via the
aforementioned ERA5 inputs, as given by the Copernicus Climate Change Service
(C3S) Climate Data Store (CDS) [23]. Furthermore, a global land-sea mask [40,
15] resampled to circa $3$ km resolution is provided. Finally, precise storm
surge measurements are derived from in situ tide gauge records collected in
GESLA-3 [9] and spatially distributed as shown in Fig. 2. Principally, our
pre-processing pipeline follows the established workflow of [37]; featuring
mean sea level de-trending, harmonic decompositioning [5] and de-noising
steps. Key differences to the prior work are their filtering of any in situ
sites with records shorter than seven years, whereas we take inspiration from
recent progress on (un)gauged river streamflow forecasting [25] and keep such
data. While shorter durations pose a greater challenge to learn site-wise
dynamics, this drastically increases the overall amount of valuable in situ
data from a total of $736$ tidal gauges in [37] to $3553$ locations in our
work — yielding an almost five-fold increase of valuable in situ data compared
to preceding efforts.

##  4 Methods

The problem tackled herein is that of forecasting the highly non-linear
dynamics of storm surges on a short lead time: Every sample
$i=0,1,...,\left|\mathcal{D}\right|$ of the dataset $\mathcal{D}$ denotes a
pair $(\bm{X^{i}},Y^{i})$, with $\bm{X}^{i}=[X^{i}_{1},\cdots,X^{i}_{T}]$
being the input time series of size $\left[T\times C_{in}\times H\times
W\right]$ featuring in situ plus atmospheric reanalysis and model-based surge
data, and $Y^{i}$ is the target image of shape $\left[C_{out}\times H\times
W\right]$ at a lead time of $L$ hours. $T$ is the temporal length of the input
series, $C_{in}$ and $C_{out}$ denote the number of input and output channels,
and $H\times W$ the images’ two spatial dimensions. For convenience, the $i$
superscript is omitted in the remainder of the paper. Unless stated otherwise,
we set $T=12$, $C_{in}=5$, $C_{out}=2$, $H=W=256$ and $L=8$ hours, which has
been a common choice in prior short-term forecasting works [34]. Example data
are illustrated in Fig. 3, showcasing the time series dynamics, the diversity
in context and the presence of extreme weather events in the data.

###  4.1 Models

We demonstrate the feasibility of our implicit assimilation and densification
approach by adapting a representative variety of models. Besides highlighting
our paradigm’s effectiveness for sparse coastal observations, these
evaluations may serve as a benchmark for future research. For each considered
network, we follow the respective architecture’s best practice in terms of
hyperparameters given in the referenced literature, unless otherwise
specified. Models are:

(a) In situ time series.

(b) Input

(c) Target

Figure 3: Example data, one local sample per row. Input grouped at the left,
targets at the right. Inputs: Time series of target (blue) and context gauges
(grey), with target history only given in the hyperlocal setting. gauge
locations and surge values. ERA5 pressure at mean sea level, wind at 10 m in u
and v directions. Coarse GTSM input. Targets: Surge at target time, GTSM at
target time.

#### Conventional baselines

As a simple baseline, we consider the seasonal average surge based on historic
values at the gauge of interest and the given target time. This necessitates
historical data at the target gauge, but is expected to provide a solid
baseline for sites experiencing seasonal cyclone activity [32]. Second, we
consider the mean surge at the gauge of interest over the input time period.
While again requiring access to the gauge’s records, this would prove
beneficial whenever the surge at target time doesn’t deviate too much from the
input period’s. Third, we consider the linear extrapolation of surge time
series inputs to the target time. Finally, the global physical storm surge
predictions of GTSM forced with ERA5 data are reported and compared against
[23]. As GTSM is numerically simulated on an irregular grid [14, 6] we perform
nearest neighbor extrapolation to get coarse forecasts at any site of
interest, specifically for getting predictions at unknown test split gauges,
and then extrapolate these to the target time.

#### LSTM-based

Long short-term memory (LSTM) networks [11] are the state-of-the-art
architectures for temporal modeling of fluvial [19, 25] and coastal dynamics
[37]. Accordingly, we consider classical as well as convolutional LSTM
(ConvLSTM) [33] for storm surge forecasting. We use the architectures from
[37] and minimally adapt them for the sake of comparability to accept our more
comprehensive data, including time series of preceding gauge values. As such,
both models receive a 5x5 context window around the gauge, and predict a
single point in the future.

#### Attention-based

We consider spatio-temporal transformer models, which both principally share a
common structure of inputs and outputs as depicted in Fig. 1. First, we
evaluate a MaxVIT U-Net backbone [38] as recently proposed for weather
forecasting in [1], adapted to our problem statement. The network collates
temporal information into the channel dimension, but is conditioned on the
lead time of the target via Feature-wise Linear Modulation (FiLM) [27].
Finally, we consider the U-TAE of [7], originally proposed for panoptic
segmentation. We adjust U-TAE by introducing FiLM at each of its convolution
blocks—such that additionally to temporal embeddings at the input time points,
our adaptation of the model is conditioned to forecast at a variable target
time. Notably, the key difference between the last two models is that [38]
resolves the input time series into the channel dimension and doesn’t model
time dynamics explicitly, whereas [7] processes time series explicitly and
applies lightweight temporal attention but doesn’t explicitly model global
spatial interactions.

###  4.2 Densification

Central to our approach generalizing storm surge forecasting to previously
unseen or ungauged sites is the concept of densification. For any model that
outputs a two-dimensional storm surge forecast map $\bm{\hat{y}}_{s}$, we
implement densification via the built-in spatial parameter sharing of the
convolution operator. Specifically, we utilize 1 $\times$ 1 convolution
kernels with a stride of $1$ at the final network layer to broadcast from
sparsely populated to non-observed pixel coordinates in the spatial
dimensions. Auxiliary supervision and input data dropout are also used to
further encourage the networks to learn densification, as proposed by
Andrychowicz et al [1].

#### Auxiliary supervision

Complementarily to learning a densified forecast of the sparse in situ time
series, the networks additionally predict a forecast $\bm{\hat{y}}_{c}$ of the
coarse GTSM ocean state at the same lead time $L$, as depicted in Fig. 1. This
way, the models receive additional feedback at pixels which would otherwise
not be populated and the preceding shared layers internalize to implicitly
assimilate the sparse observations with the coarse reanalysis. To only
evaluate the coarse ocean state predictions over valid locations we mask the
loss computation with a land sea mask $\mathds{1}_{lsm}$.

#### In situ dropout

To furthermore encourage the densifying networks to predict non-trivial
outputs at unpopulated pixels within the sparse input time series we perform
data dropout. Specifically, we randomly remove in situ tide gauges from the
input with a probability $p$ but keep all sites within the target patch, such
that the network is forced to learn extrapolating to the dropped sites. We set
$p=0.25$ and include a binary validity mask $\mathds{1}_{val}$ in the network
inputs as proposed in a weather prediction context by [1].

In sum, the densifying network architectures output two maps
$\bm{\hat{y}}=[\bm{\hat{y}}_{s},\bm{\hat{y}}_{c}]$. Map $\bm{\hat{y}}_{s}$
densely predicts the sparse GESLA gauges, and $\bm{\hat{y}}_{c}$ predicts the
spatially interpolated future coarse GTSM values. Thus, they are trained via a
weighted combination of two masked L1 cost functions

$\mathcal{L}_{s}(\bm{\hat{y}}_{s},\bm{y}_{GESLA})=\frac{1}{n}\sum_{j=1}^{n}\mathds{1}_{val}(j)\cdot\|\bm{\hat{y}}_{j}-\bm{y}_{j}\|_{1}\;,$
(1)

$\mathcal{L}_{c}(\bm{\hat{y}}_{c},\bm{y}_{GTSM})=\frac{1}{n}\sum_{j=1}^{n}\mathds{1}_{lsm}(j)\cdot\|\bm{\hat{y}}_{j}-\bm{y}_{j}\|_{1}\;,$
(2)

masked via $\mathds{1}_{val}$ and $\mathds{1}_{lsm}$, resulting in the
combined loss

$\mathcal{L}(\bm{\hat{y}},\bm{y})=\mathcal{L}_{s}(\bm{\hat{y}}_{s},\bm{y}_{GESLA})+\lambda\mathcal{L}_{c}(\bm{\hat{y}}_{c},\bm{y}_{GTSM})\;.$
(3)

We set the hyperparameter $\lambda=\frac{1}{100}$, to account for the
sparseness of in situ data in comparison to the pixel-wise coarse evaluations
and to compensate for the resulting magnitudes of differences in supervision
frequency across both domains as well as their respective loss terms.

###  4.3 Lead time conditioning

To enable a flexible forecasting, accommodating for varying hours of look-
ahead predictions at inference time and via a single forward-pass, we
implement lead time conditioning via Feature-wise Linear Modulation (FiLM)
[27]. Specifically, a shallow encoder projects queried lead times $L$ into a
low-dimensional feature space and linearly modulates convolutional feature
maps via a learned scale and bias offset. We utilize lead time conditioning
for the two considered densifying networks, and have one shallow encoder per
each of their U-Net backbone’s convolutional blocks.

##  5 Experiments

#### Splits

For our experiments, we set up splits by defining holdout data in terms of
both the spatial and temporal dimensions: A globally distributed 20 % of
coastal gauges across all ocean basins are reserved for the test split, whose
temporal extent starts from April $2014$ and is thus well observed via
satellites, in the interest of follow-up works. The remaining 70 and 10 % of
locations are utilized as training and validation splits, with records ranging
from $1979$ to $2014$. This amounts to a total of $2561$, $284$ and $708$
gauges for our train, validation and test split, respectively. The resulting
map of in situ recordings (color-coded according to their splits) and
hurricane trajectories is depicted in Fig. 2. The distribution of accessible
and gauged sites is biased towards developed countries, underlining the need
for machine learning solutions to serve under-represented regions.

#### Sampling

Having assigned a subset of gauges to the holdout splits, we determine the
test dates by identifying coincidences of in situ records with storm tracks as
given by IBTrACS [18]. When a storm passes within $100$ km of a holdout tide
gauge then we set such a date as the sample’s target time. If no storm tracks
pass by, then we resort to sampling target times at outlier surge values
deviating more than 2 train split standard deviations from the train split’s
mean surge. At train time, we likewise perform outlier sampling with a
probability of $0.5$, roughly mirroring the distribution of (non) storm events
at holdout gauges in the validation and test splits. If no such sample exists
for the current gauge, then a random target date is drawn instead.

Input gaugesEvaluation gaugesContext gaugesHoldout gaugesb) Densificationa)
Hyperlocal

Figure 4: Experimental setup. Design of the densification and hyperlocal
evaluation schemes, conceptualizing their respective inputs and outputs. The
hyperlocal protocol focuses on forecasting of novel dynamics encountered at
inference time, predicting surge at holdout target (green) and context gauges
(blue) $L$ hours ahead. The generalization setup quantifies the goodness of
models to broadcast predictions to ungauged locations, i.e. unknown gauges not
contained in the input and solely used for evaluation.

###  5.1 Implementation details

To enable fast online sampling of data and efficient training of deep
networks, we represent all spatio-temporal data as netCDF files [29] via
xarray [12], either loaded directly into memory or read in parallel via dask
[30]. This is especially critical for the in situ GESLA-3 records, which are
re-processed into a compact format as part of the preprocessing pipeline and
get released with this publication.

In contrast to weather forecasting models [26, 3, 20] that process a global
context window at proximately $30$ km resolution, the networks we consider are
significantly less resource-demanding and digest local patches of ($256$ px)2
at a finer pixel-resolution of circa $3$ km to capture local variations of
surge. All data features are z-standardized via their sufficient statistics
calculated on the training split.

#### Training

For training, 1 epoch is defined by iterating over all train split gauges in a
random order. At each gauge, a Gaussian is drawn around it’s location to
randomly sample what is treated as the local patches centroid $c$. The target
date & time $t$ are drawn randomly from the current gauge’s records. Next, a
lead time $L$ is drawn randomly from $\\{0,1,2,...,12\\}$, as in [1]. The
hourly input time sequence of length $T$ is then given by the time interval of
$[t-L-T,t-L]$. Note that the random sampling of $c$, $t$ and $L$ effectively
acts as data augmentation. For auto-regressive methods we evaluate the
prediction at time $t$, whereas single-forward-pass approaches based on FiLM
are conditioned on $L$ to directly generate a forecast at time $t$.

Table 1: Experimental evaluation. We benchmark forecasting skills for $T=12$
and $L=8$ on a global holdout set of previously unseen gauges. Point-wise
storm surge predictions at known gauges are evaluated in the hyperlocal
setting (left), whereas the densification protocol (right) tests predictions
at unknown or ungauged sites. FiLM U-TAE is the most competitive approach,
followed by MaxVIT U-Net.

Model | Hyperlocal | Densification
---|---|---
|  $\downarrow$ MAE (std) |  $\downarrow$ MSE (std) |  $\uparrow$ NNSE |  $\downarrow$ MAE (std) |  $\downarrow$ MSE (std) |  $\uparrow$ NNSE
seasonal average | 0.281 (0.313) | 0.177 (0.539) | 0.424 | — | — | —
input average | 0.267 (0.295) | 0.158 (0.452) | 0.452 | — | — | —
input extrapolation | 0.182 (0.239) | 0.090 (0.342) | 0.593 | — | — | —
GTSM extrapolation [23] | — | — | — | 0.351 (0.643) | 0.536 (4.744) | 0.195
LSTM [11, 37] | 0.166 (0.282) | 0.107 (0.759) | 0.595 | — | — | —
ConvLSTM [33, 37] | 0.162 (0.267) | 0.098 (0.691) | 0.618 | — | — | —
FiLM U-TAE [27, 7] | 0.158 (0.209) | 0.069 (0.248) | 0.655 | 0.190 (0.260) | 0.104 (0.535) | 0.556
MaxVIT U-Net [38, 1] | 0.160 (0.212) | 0.070 (0.263) | 0.649 | 0.178 (0.273) | 0.106 (0.587) | 0.552

We use the ADAM optimizer [16] at a batch size of 16, with initial learning
rates tuned over magnitudes $10^{-1}$ to $10^{-4}$ for each model
individually. All networks train for 50 epochs with an exponential learning
rate decay of 0.9. Models are evaluated on the validation split each epoch and
the checkpoint with best validation loss is used for testing.

###  5.2 Evaluation

All network predictions at target time are compared against their respective
test split in situ tide gauge values. Prediction goodness is evaluated in
terms of Mean Absolute Error (MAE) as well as Mean Squared Error (MSE),
reported in units of meters and with error-wise standard deviations (std)
across the set of test split gauges denoted in brackets. Finally, we report
each method’s Normalized Nash-Sutcliffe Efficiency (NNSE) [24], which
conceptually relates to the coefficient of determination ($R^{2}$) and takes
values within $0$ and $1$. Similar to prior weather forecasting work [1] we
evaluate in two experimental setups. The concepts of both setups are depicted
in Fig. 4 and given as follows:

#### i. Hyperlocal evaluation

In this experimental paradigm, the time series of holdout gauges not trained
upon are included into the model’s inputs at test time. Therefore, the
challenge becomes to integrate previously unseen dynamics at novel locations
and to assimilate newly encountered tidal gauges at inference time.

#### ii. Densification evaluation

Predictions at previously unseen test split gauges are obtained via
densification, i.e. a model’s ability to predict at unknown locations is
quantified here. Importantly, test split gauges are not part of the input time
series and only used as targets. Note that this setup can’t be accomplished
via previous established approaches for storm surge forecasting not
implementing densification, e.g. the conventional baselines and LSTM-based
models.

##  6 Results

###  6.1 Main experiments

To demonstrate the feasibility of our problem statement and the benefits of
our curated dataset, we evaluate all considered approaches according to their
applicability in the hyperlocal and densification experimental schemes.
Outcomes are reported in Table 1. The results show that FiLM U-TAE performs
best in the hyperlocal setting, forecasting surge at newly encountered gauges
with a mean absolute accuracy of circa $16$ cm. The seasonal average and input
average predictions are more erroneous, particularly in terms of squared
error—validating the presence of storm-driven extremes in the test data.
Overall, all neural networks outperform competing approaches in the hyperlocal
setting.

In the densification experiment, FiLM U-TAE performs best, closely followed by
MaxVIT U-Net. Notably, both densifying models denote a substantial improvement
over the GTSM baseline, whose prediction goodness exhibits elevated standard
deviations across gauges.

Altogether, FiLM U-TAE tends to outperform MaxVIT U-Net, implying that
temporal self-attention is of greater benefit than visual attention for our
spatio-temporal forecasting task. To convey a better understanding of the
spatial dependency of errors, Fig. 5 analyzes our best model’s performance
across the globe. The analysis shows that forecasting in the tropics is
particularly hard, confirming that the curated benchmark provides a
challenging problem.

Figure 5: Location of holdout gauges impacts prediction performance. Average
absolute errors in meters are color-coded and binned according to each sites’
longitude and latitude coordinates, with gauge counts overlayed for each
spatial dimension. Particularly challenging regions are the Gulf of Mexico,
the Caribbean Sea and the Indian Ocean due to their extreme climate and
resulting outsized surge dynamics.

###  6.2 Ablation experiments

In order to further investigate the challenges of short-term storm surge
forecasting and explore which design choices determine the quality of
predictions, we conduct a series of ablation studies following the
densification experimental protocol. All ablations are run with the FiLM U-TAE
network, which the preceding main experiments identified as the best
performing backbone for our approach.

Table 2: Repeated Measures. Evaluation of FiLM U-TAE with varying numbers of
input time points $T$, flexibly accommodated for via temporal self-attention.
Longer inputs tend to be beneficial.

input length T |  $\downarrow$ MAE (std) |  $\downarrow$ MSE (std) |  $\uparrow$ NNSE
---|---|---|---
6 | 0.194 (0.282) | 0.115 (0.587) | 0.551
12 | 0.190 (0.260) | 0.104 (0.535) | 0.556
18 | 0.180 (0.230) | 0.085 (0.510) | 0.573
24 | 0.180 (0.230) | 0.085 (0.510) | 0.571

Table 3: Lead Time. Evaluation of FiLM U-TAE with varying lead time offset
$L$, modifiable thanks to lead time conditioning. Storm surge forecasts become
more challenging the larger $L$ gets.

lead time t |  $\downarrow$ MAE (std) |  $\downarrow$ MSE (std) |  $\uparrow$ NNSE
---|---|---|---
4 | 0.169 (0.254) | 0.093 (0.543) | 0.583
6 | 0.182 (0.269) | 0.106 (0.551) | 0.552
8 | 0.190 (0.260) | 0.104 (0.535) | 0.556
10 | 0.191 (0.273) | 0.111 (0.553) | 0.540
12 | 0.196 (0.273) | 0.113 (0.539) | 0.536

Table 4: Input ablations. Evaluation of our models with varying inputs. The
outcomes underline the relevance of each modality.

input ablation |  $\downarrow$ MAE (std) |  $\downarrow$ MSE (std) |  $\uparrow$ NNSE
---|---|---|---
full model | 0.190 (0.260) | 0.104 (0.535) | 0.556
no GTSM input | 0.207 (0.284) | 0.124 (0.543) | 0.513
no ERA5 input | 0.189 (0.273) | 0.110 (0.545) | 0.542
no data dropout | 0.217 (0.289) | 0.130 (0.539) | 0.500
no FiLM, $L=8$ fixed | 0.183 (0.273) | 0.108 (0.567) | 0.547

Table 5: Output ablations. Evaluation of FiLM U-TAE with varying output
channels. Ablations show all outputs’ significance.

output ablation |  $\downarrow$ MAE (std) |  $\downarrow$ MSE (std) |  $\uparrow$ NNSE
---|---|---|---
full model | 0.190 (0.260) | 0.104 (0.535) | 0.556
no GTSM supervision | 0.194 (0.276) | 0.114 (0.544) | 0.534
GTSM, instead of densification | 0.210 (0.246) | 0.105 (0.536) | 0.554

#### Accuracy vs. sequence length

To evaluate the effect of the number of input time points $T$ on performances,
we run FiLM U-TAE on input time series of lengths $T=6,12,18,24$ hours. Table
2 shows that longer sequences drive better forecasts, although gains saturate.
This confirms the intuition that more context and data regarding maritime
dynamics facilitates short-term forecasting, but that observations further in
the past become less informative.

#### Accuracy vs. lead time

To evaluate the effect of the lead time $L$ on performances, we perform
inference by varying $L=4,6,8,10,12$ hours ahead. Note that the lead time can
be systematically varied in a single forward pass thanks to the network being
directly conditioned on $L$. The outcomes in Table 3 validate the intuition
that longer lead times exacerbate the prediction problem, affirming its non-
linear dynamics and the challenges of medium-term forecasting as encountered
by numerical weather prediction models.

#### Ablation studies

We systematically ablate over input information and supervision extent to
investigate each element’s significance in driving the prediction goodness.
The results are reported in Tables 4 and 5, respectively. Input ablations
justify the provisioning of coarse GTSM plus ERA5 auxiliary inputs, and show
that the network can effectively translate atmospheric reanalysis to ocean
states. Furthermore, data dropout is critical for enabling densification and
the introduction of flexible lead time conditioning is beneficial. The output
ablations confirm that coarse GTSM supervision provides valuable guidance, yet
there is additional gains for the densified storm surge output. In sum, the
outcomes validate our overall approach as depicted in Fig. 1.

(a) gauge locations.

(b) GTSM prediction.

(c) VIT U-Net densification.

(d) U-TAE densification.

(e) U-TAE coarse prediction.

Figure 6: Exemplary data and densified storm surge forecasts in the
densification experimental setup. Rows: Four samples from the test split.
Columns: Sampled gauge locations. Dense surge forecasts of GTSM, MaxVit U-Net,
FiLM U-TAE and coarse auxiliary FiLM U-TAE predictions. All illustrated
outputs are in the densification setup without the target gauge provided, at a
lead time of $L=8$ hours.

#### Qualitative results

Complementary to the reported quantitative outcomes, Fig. 6 illustrates
example data and the different densification models’ forecasts. Notably, the
networks’ densified predictions show substantial differences from GTSM, as
they are driven by the assimilation of in situ tidal gauge data which GTSM
does not incorporate. Specifically, modifications are undertaken close to the
shorelines where gauges are present and surge forecasts are most relevant.
Furthermore, the densifications often differ from the auxiliary coarse
predictions, underlining the functional differences across the two kinds of
outputs and highlighting once more the importance of integrating the sparse in
situ data. Finally, the appearance of gridding in the spatio-temporal
networks’ outputs evidences the impact of the ERA5 atmospheric information,
which is integrated in the storm surge forecasts.

##  7 Conclusion

To tackle the aggravating hazard of coastal floods, we introduce a novel
dataset and framework forecasting storm surges. Our curated data makes the
posed challenge more accessible to the remote sensing community and may serve
as a benchmark to fuel future research. Our approach is influenced by recent
progress in weather forecasting, and shows that neural networks can implicitly
assimilate sparse in situ measurements with coarse weather and ocean state
reanalysis products to provide densified forecasts. In a follow-up, we’ll
explore the operational potential of our approach and replacing retrospective
reanalysis products with recently developed forecasting models. Further
directions may be the incorporation of satellite altimetry, the modeling of
impact at landfall and the translation from storm surges to predicting flood
maps. Data and code are given at
https://github.com/PatrickESA/StormSurgeCastNet.

## References

  * [1] Marcin Andrychowicz, Lasse Espeholt, Di Li, Samier Merchant, Alex Merose, Fred Zyda, Shreya Agrawal, and Nal Kalchbrenner.  Deep learning for day forecasts from sparse observations.  arXiv preprint arXiv:2306.06079, 2023.
  * [2] Emanuele Bevacqua, Michalis I Vousdoukas, Giuseppe Zappa, Kevin Hodges, Theodore G Shepherd, Douglas Maraun, Lorenzo Mentaschi, and Luc Feyen.  More meteorological events that drive compound coastal flooding are projected under climate change.  Communications Earth & Environment, 1(1):47, 2020.
  * [3] Kaifeng Bi, Lingxi Xie, Hengheng Zhang, Xin Chen, Xiaotao Gu, and Qi Tian.  Accurate medium-range global weather forecasting with 3D neural networks.  Nature, 619(7970):533–538, 2023.
  * [4] Nicolas Bruneau, Jeff Polton, Joanne Williams, and Jason Holt.  Estimation of global coastal sea level extremes using neural networks.  Environmental Research Letters, 15(7):074030, 2020.
  * [5] Daniel L Codiga.  Unified tidal analysis and prediction using the UTide Matlab functions.  2011\.
  * [6] Deltares.  Delft3D Flexible Mesh Suite.  Delft3D-FLOW, User Manual; Deltares: Delft, The Netherlands, 2018.
  * [7] Vivien Sainte Fare Garnot and Loic Landrieu.  Panoptic segmentation of satellite image time series with convolutional temporal attention networks.  In Proceedings of the IEEE International Conference on Computer Vision, 2021.
  * [8] Avantika Gori, Ning Lin, Dazhi Xi, and Kerry Emanuel.  Tropical cyclone climatology change greatly exacerbates US extreme rainfall–surge hazard.  Nature Climate Change, 12(2):171–178, 2022.
  * [9] Ivan D Haigh, Marta Marcos, Stefan A Talke, Philip L Woodworth, John R Hunter, Ben S Hague, Arne Arns, Elizabeth Bradshaw, and Philip Thompson.  GESLA version 3: A major update to the global higher-frequency sea-level dataset.  Geoscience Data Journal, 10(3):293–314, 2023.
  * [10] Hans Hersbach, Bill Bell, Paul Berrisford, Shoji Hirahara, András Horányi, Joaquín Muñoz-Sabater, Julien Nicolas, Carole Peubey, Raluca Radu, Dinand Schepers, et al.  The ERA5 global reanalysis.  Quarterly Journal of the Royal Meteorological Society, 146(730):1999–2049, 2020.
  * [11] Sepp Hochreiter and Jürgen Schmidhuber.  Long short-term memory.  Neural computation, 9(8):1735–1780, 1997.
  * [12] S. Hoyer and J. Hamman.  xarray: N-D labeled arrays and datasets in Python.  Journal of Open Research Software, 5(1), 2017.
  * [13] Eugenia Kalnay.  Atmospheric modeling, data assimilation and predictability.  Cambridge university press, 2003.
  * [14] Herman WJ Kernkamp, Arthur Van Dam, Guus S Stelling, and Erik D de Goede.  Efficient scheme for the shallow water equations on unstructured grids with application to the continental shelf.  Ocean Dynamics, 61:1175–1188, 2011.
  * [15] Georg Erich Kindermann.  Global land / water mask map with 0.3 arc sec. (10 m) resolution.  ResearchGate, 2022.  Accessed: 2024-01-01.
  * [16] Diederik P Kingma and Jimmy Ba.  ADAM: A method for stochastic optimization.  In International Conference on Learning Representations, 2015.
  * [17] Ebru Kirezci, Ian R Young, Roshanka Ranasinghe, Sanne Muis, Robert J Nicholls, Daniel Lincke, and Jochen Hinkel.  Projections of global-scale extreme sea levels and resulting episodic coastal flooding over the 21st century.  Scientific reports, 10(1):11629, 2020.
  * [18] Kenneth R Knapp, Michael C Kruk, David H Levinson, Howard J Diamond, and Charles J Neumann.  The international best track archive for climate stewardship (IBTrACS) unifying tropical cyclone data.  Bulletin of the American Meteorological Society, 91(3):363–376, 2010.
  * [19] Frederik Kratzert, Daniel Klotz, Claire Brenner, Karsten Schulz, and Mathew Herrnegger.  Rainfall–runoff modelling using long short-term memory (LSTM) networks.  Hydrology and Earth System Sciences, 22(11):6005–6022, 2018.
  * [20] Remi Lam, Alvaro Sanchez-Gonzalez, Matthew Willson, Peter Wirnsberger, Meire Fortunato, Ferran Alet, Suman Ravuri, Timo Ewalds, Zach Eaton-Rosen, Weihua Hu, et al.  Learning skillful medium-range global weather forecasting.  Science, 382(6677):1416–1421, 2023.
  * [21] Gurvan Madec and NEMO team.  NEMO ocean engine: Note du pôle de modélisation de l’institut pierre-simon laplace no 27, 2008.
  * [22] Alexandre K Magnan, Michael Oppenheimer, Matthias Garschagen, Maya K Buchanan, Virginie KE Duvat, Donald L Forbes, James D Ford, Erwin Lambert, Jan Petzold, Fabrice G Renaud, et al.  Sea level rise risks and societal adaptation benefits in low-lying coastal areas.  Scientific reports, 12(1):10677, 2022.
  * [23] Sanne Muis, Maialen Apecechea, José Álvarez, Martin Verlaan, Kun Yan, Job Dullaart, Jeroen Aerts, Trang Duong, Rosh Ranasinghe, Dewi Bars, Rein Haarsma, and Malcolm Roberts.  Global sea level change time series from 1950 to 2050 derived from reanalysis and high resolution CMIP6 climate projections.  In Copernicus Climate Change Service (C3S) Climate Data Store (CDS), 2022.
  * [24] J Eamonn Nash and Jonh V Sutcliffe.  River flow forecasting through conceptual models: Part I — a discussion of principles.  Journal of Hydrology, 10(3):282–290, 1970.
  * [25] Grey Nearing, Deborah Cohen, Vusumuzi Dube, Martin Gauch, Oren Gilon, Shaun Harrigan, Avinatan Hassidim, Frederik Kratzert, Asher Metzger, Sella Nevo, et al.  AI increases global access to reliable flood forecasts.  arXiv preprint arXiv:2307.16104, 2023.
  * [26] Jaideep Pathak, Shashank Subramanian, Peter Harrington, Sanjeev Raja, Ashesh Chattopadhyay, Morteza Mardani, Thorsten Kurth, David Hall, Zongyi Li, Kamyar Azizzadenesheli, et al.  FourCastNet: A global data-driven high-resolution weather model using adaptive Fourier neural operators.  arXiv preprint arXiv:2202.11214, 2022.
  * [27] Ethan Perez, Florian Strub, Harm De Vries, Vincent Dumoulin, and Aaron Courville.  FiLM: Visual reasoning with a general conditioning layer.  In Proceedings of the AAAI conference on artificial intelligence, volume 32, 2018.
  * [28] Claudio Persello, Jan Dirk Wegner, Ronny Hänsch, Devis Tuia, Pedram Ghamisi, Mila Koeva, and Gustau Camps-Valls.  Deep learning and Earth observation to support the Sustainable Development Goals: Current approaches, open challenges, and future opportunities.  IEEE Geoscience and Remote Sensing Magazine, 10(2):172–200, 2022.
  * [29] Russ Rew and Glenn Davis.  NetCDF: an interface for scientific data access.  IEEE computer graphics and applications, 10(4):76–82, 1990.
  * [30] Matthew Rocklin et al.  Dask: Parallel computation with blocked algorithms and task scheduling.  In Proceedings of the 14th python in science conference, volume 130, page 136. SciPy Austin, TX, 2015.
  * [31] Jeffrey D Sachs, Christian Kroll, Guillame Lafortune, Grayson Fuller, and Finn Woelm.  Sustainable Development Report 2022.  Cambridge University Press, 2022.
  * [32] Kaiyue Shan, Yanluan Lin, Pao-Shin Chu, Xiping Yu, and Fengfei Song.  Seasonal advance of intense tropical cyclones in a warming climate.  Nature, 623(7985):83–89, 2023.
  * [33] Xingjian Shi, Zhourong Chen, Hao Wang, Dit-Yan Yeung, Wai-Kin Wong, and Wang-chun Woo.  Convolutional LSTM network: A machine learning approach for precipitation nowcasting.  Advances in neural information processing systems, 28, 2015.
  * [34] Casper Kaae Sønderby, Lasse Espeholt, Jonathan Heek, Mostafa Dehghani, Avital Oliver, Tim Salimans, Shreya Agrawal, Jason Hickey, and Nal Kalchbrenner.  MetNet: A neural weather model for precipitation forecasting.  arXiv preprint arXiv:2003.12140, 2020.
  * [35] MG Sotillo, S Cailleau, P Lorente, B Levier, R Aznar, G Reffray, A Amo-Baladrón, J Chanut, M Benkiran, and E Alvarez-Fanjul.  The MyOcean IBI ocean forecast and reanalysis systems: operational products and roadmap to the future copernicus service.  Journal of Operational Oceanography, 8(1):63–79, 2015.
  * [36] Mohsen Taherkhani, Sean Vitousek, Patrick L Barnard, Neil Frazer, Tiffany R Anderson, and Charles H Fletcher.  Sea-level rise exponentially increases coastal flood frequency.  Scientific reports, 10(1):6466, 2020.
  * [37] Timothy Tiggeloven, Anaïs Couasnon, Chiem van Straaten, Sanne Muis, and Philip J Ward.  Exploring deep learning capabilities for surge predictions in coastal areas.  Scientific reports, 11(1):17224, 2021.
  * [38] Zhengzhong Tu, Hossein Talebi, Han Zhang, Feng Yang, Peyman Milanfar, Alan Bovik, and Yinxiao Li.  Maxvit: Multi-axis vision transformer.  In European conference on computer vision, pages 459–479. Springer, 2022.
  * [39] Jun Wang, Si Yi, Mengya Li, Lei Wang, and Chengcheng Song.  Effects of sea level rise, land subsidence, bathymetric change and typhoon tracks on storm flooding in the coastal areas of shanghai.  Science of the total environment, 621:228–234, 2018.
  * [40] Daniele Zanaga, Ruben Van De Kerchove, Dirk Daems, Wanda De Keersmaecker, Carsten Brockmann, Grit Kirches, Jan Wevers, Oliver Cartus, Maurizio Santoro, Steffen Fritz, et al.  ESA WorldCover 10 m 2021 v200.  2022\.


integers. In the proof we use the fact that the polynomial
$e_{k}(\mathrm{pt})$ is the top Dickson polynomial in variables
$x_{1},\dots,x_{k}$. For more details on Dickson polynomials see for example
[36].

(1) Let $D_{k-1},D_{k-2},\dots,D_{1}$ be Dickson polynomials in variables
$x_{1},\dots,x_{k-1}$ of degree $2^{k-1}-1,2^{k-1}-2,\dots,2^{k-1}-2^{k-2}$,
respectively. In particular, $D_{k-1}=e_{k-1}(\mathrm{pt})$. From [36, Prop.
1.1] we have that

$\displaystyle D(x_{k})$
$\displaystyle:=\prod_{(\alpha_{1},\dots,\alpha_{k-1})\in\mathbb{F}_{2}^{k-1}-\\{0\\}}(\alpha_{1}x_{1}+\cdots+\alpha_{k-1}x_{k-1}+x_{k})$
$\displaystyle=x_{k}^{2^{k-1}-1}+D_{1}\,x_{k}^{2^{k-2}-1}+\dots+D_{i}\,x_{k}^{2^{k-1-i}-1}+\dots+D_{k-2}\,x_{k}+D_{k-1}.$

Here $D(x_{k})$ is considered a polynomial in
$\mathbb{F}_{2}[x_{1},\dots,x_{k-1}][x_{k}]$, and furthermore
$e_{k}(\mathrm{pt})=e_{k-1}(\mathrm{pt})x_{k}D(x_{k})$.

Let $0\leq r\leq 2^{t}-1$. We compute in
$\mathbb{F}_{2}[x_{1},\dots,x_{k-1}][x_{k}]$ as follows:

$\displaystyle D(x_{k})^{2t+r}$
$\displaystyle=\Big{(}x_{k}^{2^{k-1}-1}+\dots+D_{i}\,x_{k}^{2^{k-1-i}-1}+\dots+D_{k-2}\,x_{k}+D_{k-1}\Big{)}^{2t+r}$
$\displaystyle=\Big{(}x_{k}^{2^{t}(2^{k-1}-1)}+\dots+D_{i}^{2^{t}}\,x_{k}^{2^{t}(2^{k-1-i}-1)}+\dots+D_{k-2}^{2^{t}}\,x_{k}^{2^{t}}+D_{k-1}^{2^{t}}\Big{)}\cdot$
$\displaystyle\hskip
14.0pt\Big{(}x_{k}^{2^{k-1}-1}+\dots+D_{i}\,x_{k}^{2^{k-1-i}-1}+\dots+D_{k-2}\,x_{k}+D_{k-1}\Big{)}^{r}.$
(15)

Then, the coefficient of $x_{k}^{2^{t}(2^{k-1}-1)}$ in $D(x_{k})^{2t+r}$ is
$D_{k-1}^{r}=e_{k-1}(\mathrm{pt})^{r}$, obtained as the product of
$x_{k}^{2^{t}(2^{k-1}-1)}$ from the first factor with $D_{k-1}^{r}$ from the
second factor in (7.2). Indeed, the only other candidate which might
additionally contribute to the coefficient of $x_{k}^{2^{t}(2^{k-1}-1)}$ is
the product

$D_{1}^{2^{t}}x_{k}^{2^{t}(2^{k-2}-1)}\cdot
x_{k}^{r(2^{k-1}-1)}=D_{1}^{2^{t}}x_{k}^{2^{t}(2^{k-2}-1)+r(2^{k-1}-1)}$

when

$2^{t}(2^{k-1}-1)=2^{t}(2^{k-2}-1)+r(2^{k-1}-1)\ \Longleftrightarrow\
2^{t+k-2}=r(2^{k-1}-1).$

This cannot be because $0\leq r\leq 2^{t}-1$. Consequently, the coefficient of
$x_{k}^{2^{k-1+t}+r}$ in

$e_{k}(\mathrm{pt})^{2t+r}=e_{k-1}(\mathrm{pt})^{2t+r}x_{k}^{2t+r}D(x_{k})^{2t+r}$

is equal to $e_{k-1}(\mathrm{pt})^{2t+2r}$.

From the assumption $\iota_{k-1}(m_{1},\dots,m_{k-1})\geq 2^{t}+2r$ we have
that

$e_{k-1}(\mathrm{pt})^{2t+2r}\notin(x_{1}^{m_{1}},\dots,x_{k-1}^{m_{k-1}}),$

and since $m_{k}\geq 2^{t+k-1}+r+1$ we conclude that

$e_{k}(\mathrm{pt})^{2t+r}\notin(x_{1}^{m_{1}},\dots,x_{k-1}^{m_{k-1}},x_{k}^{m_{k}}).$

Thus, $\iota_{k}(m_{1},\dots,m_{k})\geq 2^{t}+r$ as claimed.

(2) The claim follows from the previous instance of the proposition because

$2^{t+1}+r=2^{t}+r+2^{t}>2^{t}+r+r=2^{t}+2r.$

(3) The proof is by induction on $k$ for every pair of integers $(2^{t},r)$
with $1\leq r\leq 2^{t}-1$. In the case $k=1$, the assumption $m_{1}\geq
2^{t}+r+1$ implies that

$\iota_{1}(m_{1})=m_{1}-1\geq 2^{t}+r+1-1=2^{t}+r.$

Let us assume that the claim holds for $k-1\geq 1$ and every every pair of
integers $(2^{t},r)$ with $1\leq r\leq 2^{t}-1$ (the induction hypothesis).
Take $m_{i}\geq 2^{t+k-1}+r+1=2^{(t+1)+(k-1)-1}+r+1$ for all $1\leq i\leq k$.
Applying the induction hypothesis to the first $k-1$ inequalities and the pair
$(2^{t+1},r)$ we get that

$\iota_{k-1}(m_{1},\dots,m_{k})\geq 2^{t+1}+r.$

Now, the inequalities $\iota_{k-1}(m_{1},\dots,m_{k})\geq 2^{t+1}+r+1$ and
$m_{k}\geq 2^{t+k-1}+r+1$, and the previous claim of this proposition imply
that $\iota_{k}(m_{1},\dots,m_{k})\geq 2^{t}+r$. This completes the proof.

(4) Since
$e_{k}(\mathrm{pt})^{2}=\prod_{(\alpha_{1},\dots,\alpha_{k})\in\mathbb{F}_{2}^{k}-\\{0\\}}(\alpha_{1}x_{1}^{2}+\cdots+\alpha_{k}x_{k}^{2})$,
the following equivalence holds

$e_{k}(\mathrm{pt})^{2m}\in(x_{1}^{2m_{1}},\dots,x_{k}^{2m_{k}})\
\Longleftrightarrow\
e_{k}(\mathrm{pt})^{m}\in(x_{1}^{m_{1}},\dots,x_{k}^{m_{k}}).$

This equivalence implies the claim. ∎

##  8\. Proof of Theorem 2.15

Let $E$ be a Euclidean vector bundle of dimension $n$ over a compact and
connected ENR $B$, and let the integers $1\leq k\leq n$ and $j\geq 1$ be
fixed. We first prove the equality of the ideals and then a criterion for the
existence of orthogonal partitions.

###  8.1. Proof of Part (1)

We prove the equality of the ideals

$\mathcal{J}_{k}(E):=(f_{1},\dots,f_{k})=(\overline{f}_{1},\dots,\overline{f}_{k})=:\mathcal{J}_{k}^{\prime}(E)$
(16)

where

$f_{i}:=\sum_{0\leq r_{1}+\dots+r_{i}\leq
n-i+1}w_{n-i+1-(r_{1}+\dots+r_{i})}(E)\,x_{1}^{r_{1}}\cdots x_{i}^{r_{i}},$

and

$\overline{f}_{i}:=\sum_{0\leq r_{1}+\dots+r_{k}\leq
n-i+1}w_{n-i+1-(r_{1}+\dots+r_{k})}(E)\,x_{1}^{r_{1}}\cdots x_{k}^{r_{k}},$

for $1\leq i\leq k$.

To prove the equality of the ideal we first consider the polynomials

$X_{a}[b]:=\sum_{r_{1}+\dots+r_{b}=n-a+1}x_{1}^{r_{1}}\cdots x_{b}^{r_{b}}$

for $1\leq a\leq n+1$ and $1\leq b\leq k$. It is straightforward to see that
the following equality holds

$X_{a}[b+1]=X_{a}[b]+x_{b+1}\cdot X_{a+1}[b+1].$ (17)

Indeed, we have that

$X_{a}[b+1]:=\sum_{r_{1}+\dots+r_{b}+r_{b+1}=n-a+1}x_{1}^{r_{1}}\cdots
x_{b}^{r_{b}}x_{b+1}^{r_{b+1}}=\\\ \hskip
62.0pt\sum_{r_{1}+\dots+r_{b}+0=n-a+1}x_{1}^{r_{1}}\cdots
x_{b}^{r_{b}}x_{b+1}^{0}+x_{b+1}\sum_{r_{1}+\dots+r_{b+1}=n-a}x_{1}^{r_{1}}\cdots
x_{b+1}^{r_{b+1}}=\\\ \hskip 210.0ptX_{a}[b+1]=X_{a}[b]+x_{b+1}\cdot
X_{a+1}[b+1].$

Next, using induction on $\ell\geq 0$, we prove the following identity:

$X_{c+s}[c+\ell]=\sum_{c\leq b\leq
c+\ell}\Big{(}\sum_{s_{b}+\cdots+s_{c+\ell}=b-c}x_{b}^{s_{b}}\cdots
x_{c+\ell}^{s_{c+\ell}}\Big{)}X_{b+s}[b].$ (18)

In case when $\ell=0$ the equality (18) becomes the identity
$X_{c+s}[c]=X_{c+s}[c]$, an so the induction basis is verified. Now, we assume
that the equality (18) holds for the given fixed integer $\ell\geq 1$. For the
induction step we compute and use induction hypothesis as follows:

$X_{c+s}[c+\ell+1]\overset{\eqref{eq - induction
-1}}{=}X_{c+s}[c+\ell]+x_{c+\ell+1}\cdot X_{c+s+1}[c+\ell+1]\overset{\eqref{eq
- induction}}{=}\\\ \sum_{c\leq b\leq
c+\ell}\Big{(}\sum_{s_{b}+\cdots+s_{c+\ell}=b-c}x_{b}^{s_{b}}\cdots
x_{c+\ell}^{s_{c+\ell}}\Big{)}X_{b+s}[b]+x_{c+\ell+1}\cdot
X_{c+s+1}[c+\ell+1]=\\\ \sum_{c\leq b\leq
c+\ell}\Big{(}\sum_{s_{b}+\cdots+s_{c+\ell}=b-c}x_{b}^{s_{b}}\cdots
x_{c+\ell}^{s_{c+\ell}}\Big{)}X_{b+s}[b]+\\\
x_{c+\ell+1}\cdot\sum_{s_{1}+\dots+s_{c+\ell+1}=n-c-s}x_{1}^{s_{1}}\cdots
x_{c+\ell+1}^{s_{c+\ell+1}}.$

Gathering two terms on the right hand since of the previous equality under one
sum we get that

$X_{c+s}[c+\ell+1]=\sum_{c\leq b\leq
c+\ell+1}\Big{(}\sum_{s_{b}+\cdots+s_{c+\ell}=b-c}x_{b}^{s_{b}}\cdots
x_{c+\ell+1}^{s_{c+\ell+1}}\Big{)}X_{b+s}[b].$

This completes the induction and the proof of the relation (18).

We proceed with a proof of the equality (16). Observe that for $1\leq i\leq
k$:

$f_{i}=\sum_{0\leq s\leq
n-i+1}w_{s}(E)X_{s+i}[i]\qquad\text{and}\qquad\overline{f}_{i}=\sum_{0\leq
s\leq n-i+1}w_{s}(E)X_{s+i}[k],$

and in particular that $f_{k}=\overline{f}_{k}$.

Now, using the relation (18) we have that

$\overline{f}_{i}=\sum_{0\leq s\leq n-i+1}w_{s}(E)X_{s+i}[k]\overset{\eqref{eq
- induction}}{=}\\\ \sum_{0\leq s\leq n-i+1}w_{s}(E)\Big{(}\sum_{i\leq b\leq
k}\Big{(}\sum_{s_{b}+\dots+s_{k}=b-i}x_{b}^{s_{b}}\cdots
x_{k}^{s_{k}}\Big{)}X_{s+b}[b]\Big{)}=\hskip 15.0pt\\\ \hskip
25.0pt\sum_{i\leq b\leq
k}\Big{(}\sum_{s_{b}+\dots+s_{k}=b-i}x_{b}^{s_{b}}\cdots
x_{k}^{s_{k}}\Big{)}\Big{(}\sum_{0\leq s\leq
n-i+1}w_{s}(E)X_{s+b}[b]\Big{)}=\\\ \sum_{i\leq b\leq
k}\Big{(}\sum_{s_{b}+\dots+s_{k}=b-i}x_{b}^{s_{b}}\cdots
x_{k}^{s_{k}}\Big{)}f_{b}.$

In summary,

$\overline{f}_{r}=\sum_{i\leq b\leq
k}\Big{(}\sum_{s_{b}+\dots+s_{k}=b-i}x_{b}^{s_{b}}\cdots
x_{k}^{s_{k}}\Big{)}f_{b}.$ (19)

Hence,
$(\overline{f}_{1},\dots,\overline{f}_{k})\subseteq(f_{1},\dots,f_{k})$.

On the other hand, since $f_{k}=\overline{f}_{k}$ we have that
$f_{k}\in\mathcal{J}_{k}^{\prime}(E)=(\overline{f}_{1},\dots,\overline{f}_{k})$.
Now, for $1\leq r\leq k-1$ assume that
$f_{r+1},\dots,f_{k}\in\mathcal{J}_{k}^{\prime}(E)$. Then from the equality
(19) it follows that

$\overline{f}_{r}=\sum_{r\leq b\leq
k}\Big{(}\sum_{s_{b}+\dots+s_{k}=b-r}x_{b}^{s_{b}}\cdots
x_{k}^{s_{k}}\Big{)}f_{b}=\\\ f_{r}+\sum_{r+1\leq b\leq
k}\Big{(}\sum_{s_{b}+\dots+s_{k}=b-r}x_{b}^{s_{b}}\cdots
x_{k}^{s_{k}}\Big{)}f_{b},$

and consequently, by assumption, we have

$f_{r}=\overline{f}_{r}+\sum_{r+1\leq b\leq
k}\Big{(}\sum_{s_{b}+\dots+s_{k}=b-r}x_{b}^{s_{b}}\cdots
x_{k}^{s_{k}}\Big{)}f_{b}\ \in\ \mathcal{J}_{k}^{\prime}(E).$

Thus, $(\overline{f}_{1},\dots,\overline{f}_{k})\supseteq(f_{1},\dots,f_{k})$.

We have completed the proof of the equality (16).

###  8.2. Proof of Part (2)

For the second part of the theorem assume that the class $e_{k}(B)^{j}$ does
not belong to the ideal $\mathcal{J}_{k}(E)$. The proof relies on the
criterion from Theorem 3.4. In other words, it suffices to prove that

$\operatorname{e}\big{(}\big{(}B_{k}(E)/\underline{\mathbb{R}}\big{)}^{\oplus
j}\big{)}\neq 0.$

The mod $2$ Euler class of the vector bundle
$\big{(}B_{k}(E)/\underline{\mathbb{R}}\big{)}^{\oplus j}$, or in other words
the top Stiefel–Whitney class, lives in the cohomology of
$H^{*}(Y_{k}(E);\mathbb{F}_{2})$. We show that

  * –

$H^{*}(Y_{k}(E);\mathbb{F}_{2})\cong R_{k}(B)/\mathcal{J}_{k}(E)$, and that

  * –

$w_{(2^{k}-1)j}\big{(}\big{(}B_{k}(E)/\underline{\mathbb{R}}\big{)}^{\oplus
j}\big{)}=e_{k}(B)^{j}+\mathcal{J}_{k}(E)\in R_{k}(B)/\mathcal{J}_{k}(E)$.

The second claim follows from the first claim, the fact that $B_{k}(E)$ is the
restriction of $A_{k}(E)$, and related computation of
$w_{(2^{k}-1)j}\big{(}\big{(}A_{k}(E)/\underline{\mathbb{R}}\big{)}^{\oplus
j}\big{)}$ in the proof of Theorem 3.2. Thus we need to prove only the first
statement, that is to compute the cohomology ring
$H^{*}(Y_{k}(E);\mathbb{F}_{2})$.

First, we give a description of the space $Y_{k}(E)$ as a projective bundle at
the end of the tower of projective bundles

$\textstyle{Y_{k}(E)=\mathbb{P}(E_{k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{k}}$$\textstyle{\mathbb{P}(E_{k-1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{k-1}}$$\textstyle{\
\cdots\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{2}}$$\textstyle{\mathbb{P}(E_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{1}}$$\textstyle{B,}$
(20)

where $E_{1}:=E$ and $p_{1}$ is the projection. The vector bundles
$E_{2},\dots,E_{k}$ and the maps $p_{2},\dots,p_{k}$ are defined iteratively
as follows.

Let $H(E_{1})$ be the Hopf line bundle over $\mathbb{P}(E_{1})$, and reacall
that $p_{1}\colon\mathbb{P}(E_{1})\longrightarrow B$ is the projection map.
Then $H(E_{1})$ is a vector subbundle of the pull-back vector bundle
$p_{1}^{*}E_{1}$, and we set

$E_{2}:=H(E_{1})^{\perp}$

to be the orthogonal complement of $H(E_{1})$ inside $p_{1}^{*}E_{1}$. In
particular, $E_{2}$ is a $(n-1)$-dimensional vector bundle over
$\mathbb{P}(E_{1})$. Set
$p_{2}\colon\mathbb{P}(E_{2})\longrightarrow\mathbb{P}(E_{1})$ to be the
projection map.

Next, $H(E_{2})\oplus p_{1}^{*}H(E_{1})$ is a vector subbundle of the pull-
back vector bundle $(p_{2}\circ p_{1})^{*}E_{1}$, and so we define

$E_{3}:=\big{(}H(E_{2})\oplus p_{1}^{*}H(E_{1})\big{)}^{\perp},$

and $p_{3}$ to be the projection map
$\mathbb{P}(E_{3})\longrightarrow\mathbb{P}(E_{2})$.

We continue in the same way. Assume that for $1\leq i\leq k-1$, all the vector
bundles $E_{1},\dots,E_{i}$, of dimensions $n,n-1,\dots,n-i+1$, respectively,
and the projection maps $p_{1},\dots,p_{i}$ are defined. Notice that

$H(E_{i})\oplus p_{i}^{*}H(E_{i-1})\oplus(p_{i}\circ
p_{i-1})^{*}H(E_{i-1})\oplus\cdots\oplus(p_{i}\circ\cdots\circ
p_{1})^{*}H(E_{1})$

is a vector subbundle of $(p_{i}\circ\cdots\circ p_{1})^{*}E_{1}$. We define
the vector bundle $E_{i+1}$ as the orthogonal complement

$E_{i+1}:=\Big{(}H(E_{i})\oplus
p_{i}^{*}H(E_{i-1})\oplus\cdots\oplus(p_{i}\circ\cdots\circ
p_{1})^{*}H(E_{1})\Big{)}^{\perp}.$ (21)

The map $p_{i+1}$ is defined to be the standard projection
$\mathbb{P}(E_{i+1})\longrightarrow\mathbb{P}(E_{i})$. It is clear that
$Y_{k}(E)=\mathbb{P}(E_{k})$.

Now, we use the tower of projective bundles (20), Lemma 4.1, as well as the
proof of Claim 4.2, to describe the cohomology ring
$H^{*}(Y_{k}(E);\mathbb{F}_{2})=H^{*}(\mathbb{P}(E_{k});\mathbb{F}_{2})$.

Since $H^{*}(Y_{k}(E);\mathbb{F}_{2})=H^{*}(\mathbb{P}(E_{k});\mathbb{F}_{2})$
where $\mathbb{P}(E_{k})$ is the projective bundle of the
$(n-k+1)$-dimensional vector bundle $E_{k}$ over $\mathbb{P}(E_{k-1})$ from
Lemma 4.1 we have that

$H^{*}(Y_{k}(E);\mathbb{F}_{2})\cong
H^{*}(\mathbb{P}(E_{k-1});\mathbb{F}_{2})[x_{k}]/\Big{(}\sum_{s=0}^{n-k+1}w_{n-k+1-s}(E_{k})\,x_{k}^{s}\Big{)},$

where $x_{k}$ corresponds to mod $2$ Euler class of the Hopf line bundle
$H(E_{k})$. Continuing to apply Lemma 4.1 for projective bundles
$\mathbb{P}(E_{k-1}),\dots,\mathbb{P}(E_{1})$ we get the following conclusion

$H^{*}(Y_{k}(E);\mathbb{F}_{2})\cong\\\
H^{*}(B;\mathbb{F}_{2})[x_{1},\dots,x_{k}]/\Big{(}\sum_{s=0}^{n}w_{n-s}(E_{1})\,x_{1}^{s},\dots,\sum_{s=0}^{n-k+1}w_{n-k+1-s}(E_{k})\,x_{k}^{s}\Big{)}.$
(22)

Here $x_{i}$, for all $1\leq i\leq k$, with a bit of abuse of notation,
corresponds to the mod $2$ Euler class of the Hopf line bundle $H(E_{i})$, or
more precisely to the mod $2$ Euler class of the pull-back line bundle
$(p_{k}\circ\cdots\circ p_{i+1})^{*}H(E_{i})$. Set
$f_{i}:=\sum_{s=0}^{n-i+1}w_{n-i+1-s}(E_{i})\,x_{i}^{s}$ for $1\leq i\leq k$.
Then

$H^{*}(Y_{k}(E);\mathbb{F}_{2})\cong
H^{*}(B;\mathbb{F}_{2})[x_{1},\dots,x_{k}]/\big{(}f_{1},\dots,f_{k}\big{)}.$

Now we focus on identification of Stiefel–Whitney classes of the vector
bundles $E_{1},\dots,E_{k}$ in terms of the Stiefel–Whitney classes $E$. Note
that $E_{1}=E$ by definition, and so $w(E_{1})=w(E)$. Next, from the
definition (21) of the vector bundles $E_{i}$ for $2\leq i\leq k$, as an
orthogonal complements, we get that

$\displaystyle w(E_{i})$ $\displaystyle=w\Big{(}-\big{(}H(E_{i-1})\oplus
p_{i-1}^{*}H(E_{i-2})\oplus\cdots\oplus(p_{i-1}\circ\cdots\circ
p_{1})^{*}H(E_{1})\big{)}\Big{)}$
$\displaystyle=w\big{(}-H(E_{i-1})\big{)}\cdot
w\big{(}-p_{i-1}^{*}H(E_{i-2})\big{)}\cdots w\big{(}-(p_{i-1}\circ\cdots\circ
p_{1})^{*}H(E_{1})\big{)}.$

From Lemma 4.1 we also know that

$w(H(E_{i-1}))=1+x_{i-1},\ \dots\ ,w((p_{i-1}\circ\cdots\circ
p_{1})^{*}H(E_{1}))=1+x_{1}.$

Here we assume the expected identifications of classes $x_{1},\cdots,x_{i-1}$
along the sequence of isomorphisms given in Lemma 4.1. Combining last two
observations we have that

$w(E_{i})=\frac{1}{1+x_{i-1}}\cdot\frac{1}{1+x_{i-2}}\cdots\frac{1}{1+x_{1}}=\sum_{r_{i-1}\geq
0}x_{i-1}^{r_{i-1}}\cdot\sum_{r_{i-2}\geq 0}x_{i-2}^{r_{i-2}}\ \cdots\
\sum_{r_{1}\geq 0}x_{1}^{r_{1}},$

for $2\leq i\leq k$. Consequently, we have that

$f_{i}=\sum_{s=0}^{n-i+1}w_{n-i+1-s}(E_{i})\,x_{i}^{s}=\\\ \sum_{0\leq
r_{1}+\dots+r_{i}\leq
n-i+1}w_{n-i+1-(r_{1}+\dots+r_{i})}(E)\,x_{1}^{r_{1}}\cdots x_{i}^{r_{i}}$

for every $1\leq i\leq k$.

This finishes the proof of the second claim, and so the proof of Theorem 3.4
is complete.

##  9\. Even more main results

In this section, we use methods developed in previous sections to give new
proofs and generalise results of Larry Guth & Nets Hawk Katz [20], Blagojević,
Dimitrijević Blagojević & Günter M. Ziegler [5], Schnider [31] and Soberón &
Yuki Takahashi [34].

Throughout this section $B$ will be a compact, connected ENR, and $E$ will be
a Euclidean real vector bundle of dimension $n$ over $B$. For an integer
$k\geq 1$, $E(1),\dots,E(k)$ will be finite-dimensional non-zero real vector
bundles over $B$ with $\dim E(i)=n_{i}$. As before, we write $S(E(i))$ for the
sphere bundle of $E(i)$ with fibre at $b\in B$ the space of oriented
$1$-dimensional subspacesof $E(i)_{b}$. Equivalently, $S(E(i))$ is the unit
sphere bundle for a chosen Euclidean structure. Also, we shall use $V$ for a
Euclidean vector space $V$, and sometimes see it as a vector bundle over a
point.

Recall that $A_{k}(E(1),\ldots,E(k))$ is the $2^{k}$-dimensional real vector
bundle over $\mathbb{P}(E(1))\times_{B}\cdots\times_{B}\mathbb{P}(E(k))$ with
fibre at $(L_{1},\ldots,L_{k})$, where $L_{i}\in\mathbb{P}(E(i)_{b})$, $b\in
B$, the real vector space of all functions $S(L_{1})\times\cdots\times
S(L_{k})\longrightarrow\mathbb{R}$. As a space of real-valued functions, each
fibre of $A_{k}(E(1),\ldots,E(k))$ can be equipped with a partial order by
setting

$f_{1}\leq f_{2}\quad\Longleftrightarrow\quad(\forall x\in
S(L_{1})\times\cdots\times S(L_{k}))\ f_{1}(x)\leq f_{2}(x)$

for $f_{1},f_{2}\in A_{k}(E(1),\ldots,E(k))$. Hence, every finite non-empty
subset of functions $S$ has a least upper bound, which we shall denote by
$\max(S)$.

###  9.1. Partitioning by polynomials

Now we give an extension of the results [20, Thm. 4.1], [19, Thm. 0.3] and [5,
Thm. 1.3] to the setting of mass assignments over an arbitrary real vector
bunlde $E$. In the case of a vector bundle over the point we recover the
original results.

For an integer $d\geq 0$, let $\mathcal{P}^{d}(E)$ denote the real vector
bundle of dimension $\binom{n+d-1}{d}$ over $B$ with fibre at $b\in B$ the
vector space of homogeneous polynomial functions $v\colon
E_{b}\longrightarrow\mathbb{R}$ of degree $d$. It is the dual $(S^{d}E)^{*}$
of the vector bundle obtained from of the $d$th symmetric power of $E$. If
$d=1$, we can identify $\mathcal{P}^{1}(E)=E^{*}$ with $E$ using the inner
product.

In the following the crucial property of polynomial functions that we shall
need is that for a non-zero homogeneous polynomial function
$v\in\mathcal{P}^{d}(V)$, the zero-set

$Z(v)=\\{x\in S(V)\,|\,v(x)=0\\}$

is null with respect to the Lebesgue measure on the Riemannian manifold
$S(V)$. It follows that, for any $\epsilon>0$, there is an open neighbourhood
of $Z(v)$ in the sphere $S(V)$ with volume less than $\epsilon$, consult [35].

Now we extend our discussion from Section 3.3. Assume that
$E(i)\subseteq\mathcal{P}^{d(i)}(E)$ is a vector subbundle of the vector
bundle of homogeneous polynomial functions of degree $d(i)\geq 1$. For $b\in
B$,
$(L_{1},\ldots,L_{k})\in\mathbb{P}(E(1)_{b})\times\cdots\times\mathbb{P}(E(k)_{b})$,
and $(v_{1},\dots,v_{k})\in S(L_{1})\times\cdots\times S(L_{k})$, let us
define an analogue of an orthant by

$\mathcal{A}_{b;v_{1},\ldots,v_{k}}:=\\{u\in
S(E_{b})\,|\,v_{1}(u)>0,\,\ldots,\,v_{k}(u)>0\\}.$

We note that any real continuous function on the sphere bundle $\varphi\colon
S(E)\longrightarrow\mathbb{R}$ restricts to a function $\varphi_{b}\colon
S(E_{b})\longrightarrow\mathbb{R}$ which can be integrated over the set
$\mathcal{A}_{b;v_{1},\ldots,v_{k}}$.

The first generalization of [20, Thm. 4.1], and also at the same time
extension of our Theorem 2.2, can be stated as follows.

######  Theorem 9.1.

Under the hypotheses in the text, for an integer $j\geq 1$, given continuous
functions $\varphi_{1},\dots,\varphi_{j}\colon S(E)\longrightarrow\mathbb{R}$
assume that the $\mathbb{F}_{2}$-cohomology Euler class

$e(A_{k}(E(1),\ldots,E(k))/\underline{\mathbb{R}})^{j}\in
H^{(2^{k}-1)j}(\mathbb{P}(E(1))\times_{B}\cdots\times_{B}\mathbb{P}(E(k));\mathbb{F}_{2})$

of the vector bundle
$\underline{\mathbb{R}}^{j}\otimes(A_{k}(E(1),\ldots,E(k))/\underline{\mathbb{R}})\cong(A_{k}(E(1),\ldots,E(k))/\underline{\mathbb{R}})^{\oplus
j}$ is non-zero.

Then there exists a point $b\in B$ and lines $L_{i}\in\mathbb{P}(E(i)_{b})$,
$1\leq i\leq k$, such that, for each $1\leq\ell\leq j$, the function

$S(L_{1})\times\cdots\times
S(L_{k})\longrightarrow\mathbb{R},\qquad(v_{1},\ldots,v_{k})\longmapsto\int_{\mathcal{A}_{b;\,v_{1},\ldots,v_{k}}}(\varphi_{\ell})_{b}$

is constant.

###### Proof.

As in in the Section 3.3, we define for any continuous function $\varphi\colon
S(E)\longrightarrow\mathbb{R}$ a section $s_{\varphi}$ of the vector bundle
$A_{k}(E(1),\ldots,E_{k})$ by

$s_{\varphi}(b,(L_{1},\ldots,L_{k}))(v_{1},\ldots,v_{k}):=\int_{\mathcal{A}_{b;\,v_{1},\ldots,v_{k}}}\varphi_{b}.$

Continuity of $s_{\varphi}$ follows from the fact that zero sets of polynomial
functions are sets of Lebesgue measure zero on the sphere $S(V)$. The proof
then follows the pattern of arguments in the proof of Theorem 3.3 . ∎

The result remains true if the functions $\varphi_{\ell}$ are only assumed to
be integrable in an appropriate sense. Form the locally trivial bundle
$L_{B}^{1}(S(E);\mathbb{R})\longrightarrow B$ with fibre at $b\in B$ the
Banach space $L^{1}(S(E_{b});\mathbb{R})$ of all absolutely Lebesgue
integrable functions $S(E_{b})\longrightarrow\mathbb{R}$. If $\varphi$ is a
section of this Banach bundle, then we can integrate $\varphi_{b}\in
L^{1}(S(E_{b});\mathbb{R})$ and the associated section $s_{\varphi}$ is
continuous.

Next, we extend our results to probability measures as follows. Let us write
$M_{+}(S(E))\longrightarrow B$ for the locally trivial bundle with fibre at
$b\in B$ the space $M_{+}(S(E_{b}))$ of all finite Borel measures on the
sphere $S(E_{b})$, see Section 1.2. A continuous section $\mu$ of
$M_{+}(S(E))$ will be called a family of probability measures on $S(E)$ if
$\mu_{b}\in M_{+}(S(E_{b}))$ is a probability measure for each $b\in B$. In
this more general context the zero set of a polynomial function can have
positive measure.

Now, for each $b\in B$ and every
$(L_{1},\ldots,L_{k})\in\mathbb{P}(E(1)_{b})\times\cdots\times\mathbb{P}(E(k)_{b})$,
we have $2^{k}$ non-negative real numbers
$\mu_{b}(\mathcal{A}_{b;v_{1},\ldots,v_{k}})\in\mathbb{R}$,
$(v_{1},\ldots,v_{k})\in S(L_{1})\times\cdots\times S(L_{k}),$ – the measures
of generalised orthants – with sum less than or equal to $1$ (the measure of a
zero sets can be positive).

The following proposition allow us to transfer our more general setup in the
previously developed topological framework.

######  Proposition 9.2.

Assume that for an integer $j\geq 1$ there exist families of probability
measures $\mu_{1},\ldots,\mu_{j}$ on $S(E)$ with the property that, for each
$b\in B$ and every
$(L_{1},\ldots,L_{k})\in\mathbb{P}(E(1)_{b})\times\cdots\times\mathbb{P}(E(k)_{b})$,
there is $(v_{1},\ldots,v_{k})\in S(L_{1})\times\cdots\times S(L_{k})$ and
some $\ell$ such that
$(\mu_{\ell})_{b}(\mathcal{A}_{b;v_{1},\ldots,v_{k}})>1/2^{k}$.

Then the vector bundle
$\underline{\mathbb{R}}^{j}\otimes\big{(}A_{k}(E(1),\ldots,E(k))/\underline{\mathbb{R}}\big{)}\cong(A_{k}(E(1),\ldots,E(k))/\underline{\mathbb{R}})^{\oplus
j}$ has a nowhere zero section.

###### Proof.

For a fixed integer $1\leq\ell\leq j$, consider the set of points

$U_{\ell}:=\big{\\{}\,x=(b;L_{1},\ldots,L_{k})\in\mathbb{P}(E(1))\times_{B}\cdots\times_{B}\mathbb{P}(E(k)):\\\
(\exists(v_{1},\ldots,v_{k})\in S(L_{1})\times\cdots\times S(L_{k}))\ \
(\mu_{\ell})_{b}(\mathcal{A}_{b;v_{1},\ldots,v_{k}})>1/2^{k}\,\big{\\}},$

which is an open subspace of the base space
$X:=\mathbb{P}(E(1))\times_{B}\cdots\times_{B}\mathbb{P}(E(k))$. From the
assumption it follows that $U_{1},\dots,U_{j}$ forms an open cover of the base
space $X$,

Using the local triviality of the vector bundles, for every point $x\in X$ we
can manufacture a (continuous) section $s_{\ell}^{x}$ of
$A_{k}(E(1),\ldots,E(k))$ and an open neighborhood $U_{\ell}^{x}$ of $x$ such
that for each $x^{\prime}=(b^{\prime};L_{1}^{\prime},\ldots,L_{k}^{\prime})\in
X$ the following holds

  1. (i)

$s^{x}_{\ell}(x^{\prime})(v_{1}^{\prime},\ldots,v_{k}^{\prime})\in[0,1]$, for
all $(v_{1}^{\prime},\ldots,v_{k}^{\prime})\in
S(L_{1}^{\prime})\times\cdots\times S(L_{k}^{\prime})$;

  2. (ii)

if $s^{x}_{\ell}(x^{\prime})(v_{1}^{\prime},\ldots,v_{k}^{\prime})=1$, then
$(\mu_{\ell})_{b^{\prime}}(\mathcal{A}_{b^{\prime};v_{1}^{\prime},\ldots,v_{k}^{\prime}})>1/2^{k}$;

  3. (iii)

if $x^{\prime}\in U_{\ell}^{x}$, then there is some
$(v_{1}^{\prime},\ldots,v_{k}^{\prime})$ such that
$s^{x}_{\ell}(x^{\prime})(v_{1}^{\prime},\ldots,v_{k}^{\prime})=1$.

Since $X$ is compact, and $U_{1},\dots,U_{j}$ forms an open cover of $X$ it
can be refined to a compact cover $K_{1},\ldots,K_{j}$ of $X$ with the
property that $K_{\ell}\subseteq U_{\ell}$ for $1\leq\ell\leq j$.

Now, for each $\ell$, we can choose a finite subset $S_{\ell}\subseteq
U_{\ell}$ such that $K_{\ell}\subseteq\bigcup_{x\in S_{\ell}}U_{\ell}^{x}$.
This allows as to define a continuous section $s_{\ell}$ of
$A_{k}(E(1),\ldots,E(k))$ as $s_{\ell}:=\max\,\\{s^{x}_{\ell}:x\in
S_{\ell}\\}$. Here the maximum is taken with respect to the partial order on
the space of real valued functions which was introduced at the beginning of
this section. The properties (i), (ii) and (iii) ensure that at each point
$x\in K_{\ell}$ at least one of the $2^{k}$ components of $s_{\ell}(x)$ is
equal to $1$, but that not all are equal to $1$. Thus, the associated section
$\bar{s}_{\ell}$ of $A_{k}(E(1),\ldots,E(k))/\underline{\mathbb{R}}$ has no
zeros in $K_{\ell}$. The sum $(\bar{s}_{1},\ldots,\bar{s}_{j})$ is a nowhere
zero section of
$\underline{\mathbb{R}}^{j}\otimes(A_{k}(E(1),\ldots,E(k))/\underline{\mathbb{R}})$.
∎

Now a generalization of Theorem 9.1 can be stated as follows.

######  Theorem 9.3.

Under the hypotheses in the text, suppose that for an integer $j\geq 1$,
$\mu_{1},\ldots,\mu_{j}$ are families of probability measures on the sphere
bundle $S(E)$.

If the $\mathbb{F}_{2}$-cohomology Euler class

$e(A_{k}(E(1),\ldots,E(k))/\underline{\mathbb{R}})^{j}\in
H^{(2^{k}-1)j}(\mathbb{P}(E(1))\times_{B}\cdots\times_{B}\mathbb{P}(E(k);\,\mathbb{F}_{2})$

of the vector bundle
$\underline{\mathbb{R}}^{j}\otimes\big{(}A_{k}(E(1),\ldots,E(k))/\underline{\mathbb{R}}\big{)}$
is non-zero, then there exists a point $b\in B$ and lines
$L_{i}\in\mathbb{P}(E(i)_{b})$, $1\leq i\leq k$, such that, for each
$1\leq\ell\leq j$ and every $(v_{1},\ldots,v_{k})\in
S(L_{1})\times\cdots\times S(L_{k})$,

$\mu_{\ell}(\mathcal{A}_{b;v_{1},\ldots,v_{k}})\leq\frac{1}{2^{k}}.$

###### Proof.

Since the Euler class is non-zero, every section of the vector bundle has a
zero. So the assertion follows from Proposition 9.2. ∎

As an application we give a spherical version of a generalization in [5, Thm.
1.3] of [20, Thm. 4.1].

######  Corollary 9.4.

Let $V$ be a real vector space of dimension $n$. There is a constant $C_{n}$
with the property that for integers $d\geq 1$ and $j\geq 1$, and probability
measures $\mu_{1},\ldots,\mu_{j}$ on the sphere $S(V)$, there exists a non-
zero homogeneous polynomial function $v$ of degree $d$ on $S(V)$ such that for
each component $\mathcal{O}$ of the complement in $S(V)$ of the zero set of
$v$

$\mu_{1}(\mathcal{O})<C_{n}\cdot\frac{j}{d^{n-1}},\ldots,\mu_{j}(\mathcal{O})<C_{n}\cdot\frac{j}{d^{n-1}}.$

###### Proof.

Consider probability measures $\mu_{1},\ldots,\mu_{j}$ and fix an integer
$k\geq 1$. We shall apply Theorem 9.3 with $B=\mathrm{pt}$ a point,
$V=\mathbb{R}^{n}$ and $E(i)=V(i)\subseteq\mathcal{P}^{d(i)}(V)$ a vector
subspace of dimension $n_{i}\leq{n+d(i)-1\choose d(i)}$.

Let $r(i)\geq 1$ be the least positive integer such that $r(i)^{n-1}>2^{i-1}j$
and set

$d(i)=(n-1)r(i)\qquad\text{and}\qquad n_{i}=r(i)^{n-1}.$

Take $V(i)$ to be the $n_{i}=r(i)^{n-1}$ dimensional space of polynomials with
basis, in terms of the standard coordinate functions $\xi_{i}$, the monomials

$(\xi_{1}^{s_{1}}\xi_{2}^{r(i)-s_{1}})(\xi_{2}^{s_{2}}\xi_{3}^{r(i)-s_{2}})\cdots(\xi_{n-1}^{s_{n-1}}\xi_{n}^{r(i)-s_{n-1}}),$

where $0\leq s_{1},\ldots,s_{n-1}<r(i)$.

It follows from Proposition 2.13 that $\iota_{k}(n_{1},\ldots,n_{k})\geq j$.
By Theorem 9.3, there exist homogeneous polynomials $v_{1},\dots,v_{k}$ of
degree $d(1),\dots,d(k)$, respectively, such that
$\mu_{\ell}(\mathcal{A}_{\mathrm{pt};v_{1},\ldots,v_{k}})\leq 1/2^{k}$ for all
$1\leq\ell\leq j$. The product $v_{1}\cdots v_{k}$ has degree
$d_{k}=d(1)+\ldots+d(k)$ and each component of the complement of its zero-set
is contained in some $\mathcal{A}_{\mathrm{pt};v_{1},\ldots,v_{j}}$. Since

$2^{\frac{i-1}{n-1}}\cdot j^{\frac{1}{n-1}}<r(i)\leq 2\cdot
2^{\frac{i-1}{n-1}}\cdot j^{\frac{1}{n-1}},$

it follows that

$d_{k}=d(1)+\ldots+d(k)\leq(n-1)(r(1)+\dots+r(k))\\\ \leq 2(n-1)\cdot
j^{\frac{1}{n-1}}\cdot\sum_{i=1}^{k}2^{\frac{i-1}{n-1}}=2(n-1)\cdot
j^{\frac{1}{n-1}}\cdot\frac{2^{\frac{k}{n-1}}-1}{2^{\frac{1}{n-1}}-1}.$

So $d_{k}^{n-1}<C^{\prime}_{n}2^{k}j$, where
$C^{\prime}_{n}=\Big{(}\frac{2(n-1)}{2^{\frac{1}{n-1}}-1}\Big{)}^{n-1}$.

Now $(d_{k})$ is a strictly increasing sequence. If $k$ is chosen so that
$d_{k}\leq d<d_{k+1}$, then $1/2^{k+1}<C^{\prime}_{n}j/d_{k+1}^{n-1}$, and so

$\frac{1}{2^{k}}<C_{n}\cdot\frac{j}{d_{k+1}^{n-1}}\leq
C_{n}\cdot\frac{j}{d^{n-1}},$

where $C_{n}=2C^{\prime}_{n}$. We can multiply $v_{1}\cdots v_{k}$ by any non-
zero polynomial of degree $d-d_{1}\cdots d_{k}$ to produce the required
polynomial of degree $d$. ∎

###  9.2. Partitioning by affine functions

In this section we give an extension of our results on the spherical GHR
problem for the mass assignments to the broader class of partitions by caps
which are not necessarily hemispheres.

Let $V$ be an $n$-dimensional real vector space with $n\geq 2$. Using the
inner product we can identify the vector space $\mathbb{R}\oplus V$ with the
$(n+1)$-dimensional vector space of affine functions
$V\longrightarrow\mathbb{R}$ where the pair $(t,w)\in\mathbb{R}\oplus V$
determines the function $u\longmapsto t+\langle u,w\rangle$. A unit vector
$v=(t,w)\in S(\mathbb{R}\oplus V)$ decomposes the sphere $S(V)$ as the union
$S(V)=C(v)\cup C(-v)$ of two caps:

$C(v)=\\{u\in S(V):\langle u,w\rangle\geq-t\\}\qquad\text{and}\qquad
C(-v)=\\{u\in S(V):\langle u,w\rangle\leq-t\\}$

with intersection $\\{u\in S(V):\langle u,w\rangle=-t\\}$. For an illustration
see Figure 4. If $t=0$, the caps are hemispheres. If $t>\|w\|$, then
$C(-v)=\emptyset$; if $t<-\|w\|$, then $C(v)=\emptyset$. If $t=\|w\|$, then
$C(-v)$ is the single point $-w/\|w\|$, and, if $t=-\|w\|$,
$C(v)=\\{w/\|w\|\\}$. The intersection $C(v)\cap C(-v)$, if $|t|<\|w\|$ is a
sphere of dimension $n-2$ (if $n>1$, which we now assume).

Figure 4. The halfspaces defining the caps.

Now suppose that each vector bundle $E(i)$ is a subbundle of
$\underline{\mathbb{R}}\oplus E$, regarding a vector $v\in
E(i)_{b}\subseteq\mathbb{R}\oplus E_{b}$ in the fibre at $b\in B$ as an affine
linear function $E_{b}\longrightarrow\mathbb{R}$. For a point $b\in B$, a
collection of lines
$(L_{1},\ldots,L_{k})\in\mathbb{P}(E(1)_{b})\times\cdots\times\mathbb{P}(E(k)_{b})$,
and a collection of vectors $(v_{1},\ldots,v_{k})\in
S(L_{1})\times\cdots\times S(L_{k})$, we write another analogue of an orthant
by

$\mathcal{C}_{b;v_{1},\ldots,v_{k}}:=\\{u\in
S(E_{b}):v_{1}(u)>0,\,\ldots,\,v_{k}(u)>0\\}.$

The corresponding equipartition theorem is proved in the usual way by
constructing a section of the vector bundle
$\underline{\mathbb{R}}^{j}\otimes\big{(}A_{k}(E(1),\ldots,E(k))/\underline{\mathbb{R}}\big{)}$.

######  Theorem 9.5.

Under the hypotheses in the text, suppose that for an integer $j\geq 1$,
$\varphi_{1},\dots,\varphi_{j}\colon S(E)\longrightarrow\mathbb{R}$ are
continuous functions.

If the $\mathbb{F}_{2}$-cohomology Euler class

$e(A_{k}(E(1),\ldots,E(k))/\underline{\mathbb{R}})^{j}\in
H^{(2^{k}-1)j}(\mathbb{P}(E(1))\times_{B}\cdots\times_{B}\mathbb{P}(E(k);\mathbb{F}_{2})$

of the vector bundle
$\underline{\mathbb{R}}^{j}\otimes\big{(}A_{k}(E(1),\ldots,E(k))/\underline{\mathbb{R}}\big{)}$
is non-zero, then there exists a point $b\in B$ and lines
$L_{i}\in\mathbb{P}(E(i)_{b})$, $1\leq i\leq k$ such that, for each
$1\leq\ell\leq j$, the function

$S(L_{1})\times\cdots\times
S(L_{k})\longrightarrow\mathbb{R},\quad(v_{1},\ldots,v_{k})\longmapsto\int_{\mathcal{C}_{b;\,v_{1},\ldots,v_{k}}}\,(\varphi_{\ell})_{b}$

is constant.

###  9.3. Partitioning by spherical weges

Next we describe an extension of the results of Schnider [31] and Soberón &
Takahashi [34].

Let $V$ be a vector space of dimension $n\geq 3$, and let $U\subseteq V$ be a
vector subspace of dimension $m\geq 2$. Then $V=U\oplus U^{\perp}$ is the
direct sum of $U$ and its orthogonal complement $U^{\perp}$ and the unit
sphere $S(V)=S(U\oplus U^{\perp})$ is the join $S(U)*S(U^{\perp})$. To be
precise, we also think of the join as the space

$S(V)=\\{\cos(\theta)x+\sin(\theta)y\,|\,x\in S(U),\,y\in
S(U^{\perp}),\,0\leq\theta\leq\pi/2\\}.$

Like in the previous section, given $v=(t,w)\in S(\mathbb{R}\oplus U)$, we
have the decomposition of the sphere $S(U)$ as $C(v)\cup C(-v)$, where

$C(v)=\\{u\in S(U):\langle u,w\rangle\geq-t\\}\qquad\text{and}\qquad
C(v)=\\{u\in S(U):\langle u,w\rangle\geq-t\\}.$

This leads to a decomposition of the bigger sphere $S(V)$ as the union
$W(v,U)\cup W(-v,U)$ of two wedges

$\displaystyle W(v,U)$ $\displaystyle=C(v)*S(U^{\perp})$
$\displaystyle=\\{\cos(\theta)u+\sin(\theta)y:u\in S(U),\,y\in
S(U^{\perp}),\,\langle u,w\rangle\geq-t,\,0\leq\theta\leq\pi/2\\}$

and

$\displaystyle W(-v,U)$ $\displaystyle=C(-v)*S(U^{\perp})$
$\displaystyle=\\{\cos(\theta)u+\sin(\theta)y:u\in S(U),\,y\in
S(U^{\perp}),\,\langle u,w\rangle\leq-t,\,0\leq\theta\leq\pi/2\\}$

The intersection $W(v,U)\cap W(-v,U)$ is $S(U^{\perp})$ if $|t|>\|w\|$, a disc
of dimension $n-m$ if $|t|=\|w\|$, and a sphere of dimension $n-2$ if
$|t|<\|w\|$. (The subspace $\\{rx:r\geq 0,\,x\in W(v,U)\\}$ of $V$ is an
$m$-cone in the sense of [31].)

For example, take $U=\mathbb{R}^{2}$, $U^{\perp}=\mathbb{R}$,
$V=\mathbb{R}^{2}\oplus\mathbb{R}$, so that $m=2$, $n=3$. The wedges $W(v,U)$,
where $v=(t,w)\in S(\mathbb{R}^{2}\oplus\mathbb{R})$ with $|t|<\|w\|$, are the
subsets

$\\{(\cos(\theta)\cos(\phi),\,\cos(\theta)\sin(\phi),\,\sin(\theta))\in
S(\mathbb{R}^{2}\oplus\mathbb{R}):\alpha\leq\phi\leq\beta,\,-\pi/2\leq\theta\leq\pi/2\\}$

where $0\leq\alpha<\beta<2\pi$.

Now suppose that $F(i)\subseteq E$ is a vector subbundle of dimension
$m_{i}\geq 2$, for every $1\leq i\leq k$, and that $E(i)$ is a subbundle of
$\underline{\mathbb{R}}\oplus F(i)$ of dimension $n_{i}\leq m_{i}+1$. For a
point $b\in B$, lines
$(L_{1},\ldots,L_{k})\in\mathbb{P}(E(1)_{b})\times\cdots\times\mathbb{P}(E(k)_{b})$,
and vectors $(v_{1},\ldots,v_{k})\in S(L_{1})\times\cdots\times S(L_{k})$, we
write

$\mathcal{W}_{b;v_{1},\ldots,v_{k}}:=\bigcap_{i=1}^{k}\big{(}S(E_{b})-W(-v_{i},F(i)_{b})\big{)}$

as the intersection of the open subsets $S(E_{b})-W(-v_{i},F(i)_{b})\subseteq
W(v_{i},E(i)_{b})$.

As in all the previous partition problems for mass assignments we derive the
following result in almost identical manner.

######  Theorem 9.6.

Under the hypotheses in the text, suppose that $j\geq 1$ is an integer,
$\varphi_{1},\dots,\varphi_{j}\colon S(E)\longrightarrow\mathbb{R}$ are
continuous functions, and $j\leq\iota_{k}(E(1),\ldots,E(k))$.

Then there exists a point $b\in B$ and lines $L_{i}\in\mathbb{P}(E(i)_{b})$,
$1\leq i\leq k$, such that, for each $1\leq\ell\leq j$, the function

$S(L_{1})\times\cdots\times
S(L_{k})\longrightarrow\mathbb{R},\qquad(v_{1},\ldots,v_{k})\longmapsto\int_{\mathcal{W}_{b;\,v_{1},\ldots,v_{k}}}\,(\varphi_{\ell})_{b}$

is constant.

In the special case of a vector bundle over a point we get the following
corollary.

######  Corollary 9.7.

Suppose that $j\geq 1$ is an integer,
$\varphi_{1},\ldots,\phi_{j}:S(V)\longrightarrow\mathbb{R}$ are continuous
functions and that $j\leq\iota_{k}(n+1,\ldots,n+1)$. Let $m_{1},\ldots,m_{k}$
be integers in the range $2\leq m_{i}\leq n$.

Then there exist vector subspaces $U_{1},\ldots,U_{k}\subseteq V$ with
$\dim(U_{i})=m_{i}$ and lines $L_{i}\in\mathbb{P}(\mathbb{R}\oplus U_{i})$,
$1\leq i\leq k$, such that for each $1\leq\ell\leq j$, the function

$S(L_{1})\times\cdots\times
S(L_{k})\longrightarrow\mathbb{R},\quad(v_{1},\ldots,v_{k})\longmapsto\int_{W(v_{1},U_{1})\,\cap\,\cdots\,\,\cap\,W(v_{k},U_{k})}\,\varphi_{\ell}$

is constant.

###### Proof.

Take $B$ to be the product $G_{m_{1}}(V)\times\cdots\times G_{m_{k}}(V)$ of
Grassmann manifolds and $F(i)$ to be the canonical $m_{i}$-dimensional bundle
over the $i$th factor. Apply Theorem 9.6 with $n_{i}=m_{i}+1$ and
$E(i)=\underline{\mathbb{R}}\oplus F(i)$. Indeed, since
$\iota_{1}(\underline{\mathbb{R}}\oplus F(i))=n$, we have that
$\iota_{k}(E(1),\ldots,E(k))=\iota_{k}(n+1,\ldots,n+1)$ by Proposition 2.7. ∎

######  Remark 9.8.

The previous Corollary 9.7 can be sharpened by restricting the base space in
the following way. Replace the Grassmann manifolds $G_{m_{i}}(V)$, where
$V=\mathbb{R}^{n}$, by the its subspace $\mathbb{P}(\mathbb{R}^{n-m_{i}+1})$,
embedded by taking the direct sum of a line in $\mathbb{R}^{n-m_{i}+1}$ with
$\mathbb{R}^{m_{i}-1}$ to get a subspace of
$\mathbb{R}^{n}=\mathbb{R}^{n-m_{i}+1}\oplus\mathbb{R}^{m_{i}-1}$ of dimension
$m_{i}$. Then the vector bundle $E(i)$ restricts to
$\underline{\mathbb{R}}^{m_{i}}\oplus H_{i}$ where $H_{i}$ is the Hopf line
bundle $H(\mathbb{R}^{n-m_{i}+1})$. We have
$\iota_{1}(\underline{\mathbb{R}}^{m_{i}}\oplus H_{i})=n$, because
$w_{n-m_{i}}(-H_{i})\not=0$.

To illustrate the given in Corollary 9.7 we spell out the special case $n=3$,
$j=3$, $k=1$, $m_{1}=2$, for which $\iota_{1}(3+1)=3$. Suppose that
$\varphi_{1},\varphi_{2},\varphi_{3}\colon
S^{2}=S(\mathbb{R}^{3})\longrightarrow\mathbb{R}$ are continuous functions.
Then there is a wedge $W\subseteq\mathbb{R}^{3}$, specified by a plane $U$
through the origin in $\mathbb{R}^{3}$ and $(t,w)\in S(\mathbb{R}\oplus U)$,
such that

$\int_{W}\phi_{1}=\frac{1}{2}\int_{S^{2}}\phi_{1}\,,\qquad\int_{W}\phi_{2}=\frac{1}{2}\int_{S^{2}}\phi_{2}\,,\qquad\int_{W}\phi_{3}=\frac{1}{2}\int_{S^{2}}\phi_{3}\,.$

Furthermore, the $k=1$ case of Corollary 9.7 gives [31, Thm. 8], and also the
spherical version of [34, Thm. 1.2 and Thm. 3.2].

######  Corollary 9.9.

Suppose that $\varphi_{1},\ldots,\varphi_{n}\colon
S(\mathbb{R}^{n})\longrightarrow\mathbb{R}$ are continuous functions and $m$
is an integer, $2\leq m\leq n$. Write $V=\mathbb{R}^{n}$ and
$V^{\prime}=\mathbb{R}^{m-1}\subseteq\mathbb{R}^{n-m+1}\oplus\mathbb{R}^{m-1}=V$.

Then there exists a vector subspace $U\subseteq V$ of dimension $m$ containing
the subspace $V^{\prime}$ and a vector $v\in S(\mathbb{R}\oplus U)$ such that

$\int_{W(v,U)}\varphi_{\ell}=\frac{1}{2}\int_{S(V)}\phi_{l}=\int_{W(-v,U)}\varphi_{\ell}$

for $\ell=1,\ldots,n$.

###### Proof.

We just need to recall that $\iota_{1}(n+1)=n$. The sharpening, to give the
restriction that $U$ should contain $V^{\prime}$, is given by Remark 9.8. ∎

The connection between the affine and spherical cases was discussed in Section
1.3. We explain how [34, Thm. 1.2] can be deduced from the case $m=2$ of our
Corollary 9.9.

######  Corollary 9.10.

For an integer $n\geq 2$, suppose that
$\psi_{1},\ldots,\psi_{n}\colon\mathbb{R}^{n-1}\longrightarrow\mathbb{R}$ are
continuous functions with compact support with the $n$ integrals
$\int_{\mathbb{R}^{n-1}}\psi_{\ell}$, $1\leq\ell\leq n$, not all equal to
zero.

Then there exist two distinct parallel hyperplanes in $\mathbb{R}^{n-1}$ such
that the closed region $S$ sandwiched between them satisfies

$\int_{S}\psi_{l}=\frac{1}{2}\int_{\mathbb{R}^{n-1}}\psi_{\ell}\,,$

for all $1\leq\ell\leq n$.

(Note that if all the integrals $\int_{\mathbb{R}^{n-1}}\phi_{\ell}$ are zero,
then there is a trivial statement for any two coinciding hyperplanes.)

###### Proof.

Consider the diffeomorphism

$\pi\colon\Lambda=\\{(x,y)\in
S(\mathbb{R}^{n-1}\oplus\mathbb{R}):y>0\\}\longrightarrow\mathbb{R}^{n-1}\,,\qquad\qquad(x,y)\longmapsto\frac{x}{y}\,,$

which maps intersections of linear subspaces of
$\mathbb{R}^{n-1}\oplus\mathbb{R}$ with $\Lambda$ to affine subspaces of
$\mathbb{R}^{n-1}$. Each density $\psi_{\ell}$ lifts to a density
$\varphi_{\ell}$ on $S(\mathbb{R}^{n-1}\oplus\mathbb{R})$ with support in the
open upper hemisphere $\Lambda$. (To be precise,
$\varphi(x,y)=y^{n}\psi(x/y)$.)

Let $U\subseteq\mathbb{R}^{n-1}\oplus\mathbb{R}=V$ be a $2$-dimensional vector
subspace and $v\in S(\mathbb{R}\oplus U)$ a vector as provided by Corollary
9.9 when $m=2$. Since some $\int_{\mathbb{R}^{n-1}}\psi_{\ell}$ is non-zero,
both $S(V)-W(-v,U)$ and $S(V)-W(v,U)$ have to be non-empty. The intersection
$W(v,U)\cap W(-v,U)$ is, therefore, the union of two discs

$\\{a\\}*S(U^{\perp})\subseteq S(\mathbb{R}\cdot a\oplus
U^{\perp})\qquad\text{and}\qquad\\{b\\}*S(U^{\perp})\subseteq
S(\mathbb{R}\cdot b\oplus U^{\perp})$

meeting in $S(U^{\perp})$. Here $a,b\in S(U)$.

The image of the intersection $\pi(W(v,U)\cap W(-v,U)\cap\Lambda)$ is the
union of two affine hyperplanes meeting in $\pi(S(U^{\perp})\cap\Lambda)$.

We can prescribe the subspace $V^{\prime}$ in Corollary 9.9 to be the line
$0\oplus\mathbb{R}\subseteq\mathbb{R}^{n-1}\oplus\mathbb{R}$. In that case,
$S(U^{\perp})\cap\Lambda$ is empty, and the two hyperplanes are parallel. ∎

##  10\. Concluding remarks: Real flag manifolds

In the final section we make further remarks on particular arguments used in
the proofs of our results.

For a Euclidean vector space $V$ of dimension $n$ and integers
$0=n_{0}<n_{1}<\cdots<n_{k}<n$, let
$B:=\operatorname{Flag}_{n_{1},\dots,n_{k}}(V)$ be the manifold of flags
$(V_{*}):0=V_{0}\subseteq V_{1}\subseteq\cdots\subseteq V_{k}\subseteq V$ with
$\dim V_{i}=n_{i}$. The canonical bundles of dimension $n_{i}$ over $B$ are
denoted by $E(i)$, as in the statement of Corollary 2.10. Write $E$ for the
trivial bundle over $B$ with fibre $V$.

######  Proposition 10.1.

The $\mathbb{F}_{2}$-Euler classes satisfy

$\prod_{i=1}^{k}\operatorname{e}(E/E(i))^{n_{i}-n_{i-1}}\not=0\in
H^{d}(B;\,\mathbb{F}_{2})=\mathbb{F}_{2},$

where $n_{0}=0$ and the dimension $d$ is equal to
$\sum_{i=1}^{k}(n-n_{i})(n_{i}-n_{i-1})$.

###### Proof.

Let $(U_{*}):0=U_{0}\subseteq U_{1}\subseteq\cdots\subseteq U_{k}$ be a fixed
flag in $V$. The general linear group $G={\rm GL}(V)$ acts transitively on
$B$. If $H\leq G$ denotes the stabilizer of $(U_{*})$, we have a map
$\pi\colon G\longrightarrow B$ defined given $\pi(g)=(gU_{*})$ which describes
$B$ as the homogeneous space $G/H$.

The derivative of $\pi$ at $1\in G$ is a map from the Lie algebra
$\mathfrak{g}={\rm End}(V)$ onto the tangent space of $B$ at $(U_{*})$ with
kernel the Lie algebra $\mathfrak{h}$ of $H$, that is, the space of
endomorphisms $a$ of $V$ such that $a(U_{i})\subseteq U_{i}$ for all $i$. The
tangent bundle of $B$ is the quotient of the trivial Lie algebra bundle
$B\times{\rm GL}(V)={\rm GL}(F)$ by the subbundle with fibre at $(V_{*})\in B$
the quotient of ${\rm End}(V)$ by the Lie subalgebra, $\mathfrak{h}(V_{*})$ of
endomorphisms that preserve the flag. Using the inner product, we can express
$\mathfrak{h}(V_{*})$ as $\bigoplus_{i=1}^{k}{\rm
Hom}(V_{i}^{\perp},V_{i}\cap(V_{i-1}^{\perp}))$, which has dimension
$\sum_{i=1}^{k}(n-n_{i})(n_{i}-n_{i-1})$.

Now consider the vector bundle $E^{\prime}$, defined as a quotient of
$B\times{\rm End}(V)$, with the fibre at $(V_{*})\in B$ the quotient of
$\mathfrak{g}={\rm End}(V)$ by the vector subspace $\mathfrak{h}(V_{*},U_{*})$
of maps $a\colon V\longrightarrow V$ such that $a(V_{i})\subseteq U_{i}$ for
$i=1,\ldots,k$. In metric terms,

$E^{\prime}=\bigoplus_{i=1}^{k}{\rm
Hom}(F(i)^{\perp},U_{i}\cap(U_{i-1}^{\perp}))\,,$

and its Euler class $\operatorname{e}(E^{\prime})$ is equal to
$\prod_{i=1}^{k}e(E(i)^{\perp})^{n_{i}-n_{i-1}}\in H^{d}(B;\,\mathbb{F}_{2})$.

The vector bundle $E^{\prime}$ over the closed connected $d$-dimensional
manifold $B$ has the same dimension $d$. We shall prove that
$\operatorname{e}(E^{\prime})$ is non-zero by writing down a smooth section
$s$ of $E^{\prime}$ with exactly one zero and checking that the (mod $2$)
degree of that zero is equal to $1$.

The section $s$ is defined to have the value at $(V_{*})$ given, modulo
$\mathfrak{h}(V_{*},U_{*})$, by the identity endomorphism $1\in{\rm End}(V)$.
At a zero of $s$, $V_{i}\subseteq U_{i}$ for all $i$, that is, $V_{*}=U_{*}$.
At this zero, the tangent space of $B$ coincides with the fibre of
$E^{\prime}$, and we shall show that the derivative of $s$ is the identity
endomorphism of $\mathfrak{g}/\mathfrak{h}$.

To do this, we lift from $B=G/H$ to $G$ by the projection $\pi$. The pullback
$\pi^{*}E^{\prime}$ is trivialized by the isomorphism

$G\times(\mathfrak{g}/\mathfrak{h})\longrightarrow E^{\prime}$

taking $(g,a+\mathfrak{h})$, where $g\in{\rm GL}(V)$, $a\in{\rm End}(V)$, to
$((gU_{*}),ag^{-1}+\mathfrak{h}(gU_{*}))$. And the section $s$ lifts to the
map

$G\to\mathfrak{g}/\mathfrak{h}\,:\,g\longmapsto g+\mathfrak{h},$

for which the derivative at $1$ is, transparently, the projection
$\mathfrak{g}\to\mathfrak{g}/\mathfrak{h}$. This completes the proof. ∎

Writing the quotient $E/E(i)=E(i)^{\perp}$ as the direct sum
$\bigoplus_{j=i}^{k}(E(j+1)/E(j))$, where $E(k+1)=E$, we can reformulate
Proposition 10.1 as follows.

######  Corollary 10.2.

The product of Euler classes

$\prod_{i=1}^{k}\operatorname{e}(E(i+1)/E(i))^{n_{i}}\in
H^{d}(B;\,\mathbb{F}_{2})$

is non-zero.

The previous Corollary 10.2 connects with previously given arguments in the
following way:

  * –

The case $k=1$, shows that $\operatorname{e}(E(1)^{\perp})^{n-n_{1}}\not=0$,
and in particular $\operatorname{e}(E(1)^{\perp})\not=0$, as used in the proof
of Corollary 2.5.

  * –

The statement $\operatorname{e}(E(1)^{\perp})^{n-n_{1}}\not=0$ is the result
needed in Section 5.3 for the proof of Corollary 2.8.

  * –

For general $k$, we have in particular that
$\prod_{i=1}^{k}\operatorname{e}(E(i)^{\perp})\not=0$. This is what is
required in Section 6.1 to prove Corollary 2.10.

  * –

If $n_{i}=n-k+i-1$ (that is, $n_{1}=n-k$, $n_{2}=n-k+1$, $\ldots$,
$n_{k}=n-1$), then
$\operatorname{e}(E(1)^{\perp})^{k}\operatorname{e}(E(2)^{\perp})^{k-1}\cdots\operatorname{e}(E(k)^{\perp})^{1}\not=0$.
This is what is needed in Section 6.3 to prove Theorem 2.12. It shows directly
that $\operatorname{e}(E_{k+1})^{k}\cdots\operatorname{e}(E_{d+1})^{d}\not=0$.
The permutation symmetry of the cohomology then gives

$\operatorname{e}(E_{k+1})^{j_{k}}\cdots\operatorname{e}(E_{d+1})^{j_{d}}\not=0.$

Thus, different arguments we offered in the proofs can be seen as a direct
consequences of Corollary 10.2. To best of our knowledge these implications
were not known until now, and we believe it was worth explaining these
connections.

### Acknowledgements.

The authors would like to thank Aleksandra Dimitrijević Blagojević and Matija
Blagojević for useful discussions and many improvements of the manuscript.

## References

  * [1] David Avis, _Nonpartitionable point sets_ , Inform. Process. Lett. 19 (1984), no. 3, 125–129.
  * [2] Ilani Axelrod-Freed and Pablo Soberón, _Bisections of mass assignments using flags of affine spaces_ , Discrete Comput. Geom. (2022), 19pp, Online first, 30 December 2022.
  * [3] I. Bárány and J. Matoušek, _Simultaneous partitions of measures by $k$-fans_, Discrete Comput. Geom. 25 (2001), no. 3, 317–334.
  * [4] William A. Beyer and Andrew Zardecki, _The early history of the ham sandwich theorem_ , Amer. Math. Monthly 111 (2004), no. 1, 58–61.
  * [5] Pavle V. M. Blagojević, Aleksandra S. Dimitrijević Blagojević, and Günter M. Ziegler, _Polynomial partitioning for several sets of varieties_ , J. Fixed Point Theory Appl. 19 (2017), no. 3, 1653–1660.
  * [6] Pavle V. M. Blagojević, Jaime Calles Loperena, Michael Crabb, and Aleksandra S. Dimitrijević Blagojević, _Topology of the Grünbaum–Hadwiger–Ramos problem for mass assignments_ , Topol. Methods Nonlinear Anal., to appear, 22 pages, 2022, arXiv:2208.00666.
  * [7] Pavle V. M. Blagojević, Florian Frick, Albert Haase, and Günter M. Ziegler, _Hyperplane mass partitions via relative equivariant obstruction theory_ , Doc. Math. 21 (2016), 735–771.
  * [8] by same author, _Topology of the Grünbaum-Hadwiger-Ramos hyperplane mass partition problem_ , Trans. Amer. Math. Soc. 370 (2018), no. 10, 6795–6824.
  * [9] Pavle V. M. Blagojević and Roman Karasev, _Extensions of theorems of Rattray and Makeev_ , Topol. Methods Nonlinear Anal. 40 (2012), no. 1, 189–213.
  * [10] Pavle V. M. Blagojević and Günter M. Ziegler, _The ideal-valued index for a dihedral group action, and mass partition by two hyperplanes_ , Topology Appl. 158 (2011), no. 12, 1326–1351.
  * [11] Armand Borel, _La cohomologie mod $2$ de certains espaces homogènes_, Comment. Math. Helv. 27 (1953), 165–197.
  * [12] Glen E. Bredon, _Introduction to compact transformation groups_ , Academic Press, New York-London, 1972, Pure and Applied Mathematics, Vol. 46.
  * [13] Theodor Bröcker and Tammo tom Dieck, _Kobordismentheorie_ , Lecture Notes in Mathematics, Vol. 178, Springer-Verlag, Berlin-New York, 1970.
  * [14] Michael Crabb, _Borsuk–ulam theorems for elementary abelian $2$-groups_, Topol. Methods Nonlinear Anal., to appear, 10 pages, 2022, arXiv:2201.09564.
  * [15] Michael C. Crabb, _Bisection of measures on spheres and a fixed point theorem_ , Topological Methods in Nonlinear Analysis (2021), 1 – 16.
  * [16] Michael C. Crabb and Jan Jaworowski, _Aspects of the Borsuk-Ulam theorem_ , J. Fixed Point Theory Appl. 13 (2013), no. 2, 459–488.
  * [17] Albrecht Dold, _Parametrized Borsuk–Ulam theorems_ , Commentarii Mathematici Helvetici 63 (1988), no. 1, 275–285.
  * [18] Branko Grünbaum, _Partitions of mass-distributions and of convex bodies by hyperplanes_ , Pacific J. Math. 10 (1960), 1257–1261.
  * [19] Larry Guth, _Polynomial partitioning for a set of varieties_ , Math. Proc. Cambridge Philos. Soc. 159 (2015), no. 3, 459–469.
  * [20] Larry Guth and Nets Hawk Katz, _On the Erdős distinct distances problem in the plane_ , Ann. of Math. (2) 181 (2015), no. 1, 155–190.
  * [21] Hugo Hadwiger, _Simultane Vierteilung zweier Körper_ , Arch. Math. (Basel) 17 (1966), 274–278.
  * [22] Jean-Claude Hausmann, _Mod two homology and cohomology_ , Universitext, Springer, Cham, 2014.
  * [23] Howard L. Hiller, _On the cohomology of real Grassmanians_ , Trans. Amer. Math. Soc. 257 (1980), no. 2, 521–533.
  * [24] Dale Husemoller, _Fibre bundles_ , third ed., Graduate Texts in Mathematics, vol. 20, Springer-Verlag, New York, 1994.
  * [25] Peter Mani-Levitska, Siniša Vrećica, and Rade Živaljević, _Topology and combinatorics of partitions of masses by hyperplanes_ , Adv. Math. 207 (2006), no. 1, 266–296.
  * [26] Jiří Matoušek, _Using the Borsuk-Ulam theorem_ , Universitext, Springer-Verlag, Berlin, 2003, Lectures on topological methods in combinatorics and geometry, Written in cooperation with Anders Björner and Günter M. Ziegler.
  * [27] Daniel R. Mauldin (ed.), _The Scottish Book_ , Birkhäuser, Boston, Mass., 1981, Mathematics from the Scottish Café, Including selected papers presented at the Scottish Book Conference held at North Texas State University, Denton, Tex., May 1979.
  * [28] John W. Milnor and James D. Stasheff, _Characteristic classes_ , Annals of Mathematics Studies, No. 76, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974.
  * [29] Edgar A. Ramos, _Equipartition of mass distributions by hyperplanes_ , Discrete Comput. Geom. 15 (1996), no. 2, 147–167.
  * [30] Edgardo Roldán-Pensado and Pablo Soberón, _A survey of mass partitions_ , Bull. Amer. Math. Soc. (N.S.) 59 (2022), no. 2, 227–267.
  * [31] Patrick Schnider, _Equipartitions with Wedges and Cones_ , 16 pages, 2019, arXiv:1910.13352.
  * [32] by same author, _Ham-Sandwich cuts and center transversals in subspaces_ , 35th International Symposium on Computational Geometry, LIPIcs. Leibniz Int. Proc. Inform., vol. 129, Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2019, pp. Art. No. 56, 15.
  * [33] by same author, _Ham-Sandwich Cuts and Center Transversals in Subspaces_ , Discrete Comput. Geom. 64 (2020), no. 4, 1192–1209.
  * [34] Pablo Soberón and Yuki Takahashi, _Lifting methods in mass partition problems_ , 16 pages, 2021, arXiv:2109.03749.
  * [35] Hassler Whitney, _Elementary structure of real algebraic varieties_ , Ann. of Math. (2) 66 (1957), 545–556.
  * [36] Clarence Wilkerson, _A primer on the Dickson invariants_ , Proceedings of the Northwestern Homotopy Theory Conference (Evanston, Ill., 1982), Contemp. Math., vol. 19, Amer. Math. Soc., Providence, RI, 1983, pp. 421–434.