# D3A-TS: Denoising-Driven Data Augmentation in Time Series

David Solís-Martín1,2 Juan Galán-Páez1,2 and Joaquín Borrego-Díaz2 1Datrik
Ingellicence, S.A. Seville, Spain
<EMAIL_ADDRESS><EMAIL_ADDRESS>
2Department of Computer Science and Artificial Intelligence, Seville
University, Seville, Spain
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

It has been demonstrated that the amount of data is crucial in data-driven
machine learning methods. Data is always valuable, but in some tasks, it is
almost like gold. This occurs in engineering areas where data is scarce or
very expensive to obtain, such as predictive maintenance, where faults are
rare. In this context, a mechanism to generate synthetic data can be very
useful. While in fields such as Computer Vision or Natural Language Processing
synthetic data generation has been extensively explored with promising
results, in other domains such as time series it has received less attention.
This work specifically focuses on studying and analyzing the use of different
techniques for data augmentation in time series for classification and
regression problems. The proposed approach involves the use of diffusion
probabilistic models, which have recently achieved successful results in the
field of Image Processing, for data augmentation in time series. Additionally,
the use of meta-attributes to condition the data augmentation process is
investigated. The results highlight the high utility of this methodology in
creating synthetic data to train classification and regression models. To
assess the results, six different datasets from diverse domains were employed,
showcasing versatility in terms of input size and output types. Finally, an
extensive ablation study is conducted to further support the obtained
outcomes.

###### keywords:

Time Series, Data Augmentation, Deep Learning, Synthetic Data, Diffusion

††journal: To decide

##  1 Introduction

The advances in deep learning over the past few years have demonstrated that
the amount of data used for training models is crucial. The more data
available for training, the more generalized could be the model obtained;
increasing the training data set size decreases the probability of
overfitting. However, real-word data is limited [1] and the process of
collecting and labeling data can be expensive [2] [3]. This limitation makes
it impractical to obtain a large dataset, particularly in cases such as
predictive maintenance where failure events occur infrequently. In such
contexts, having a mechanism to expand the available data becomes crucial for
improving the generalization of models.

One tool for increasing both the quantity and quality of data is known as data
augmentation. Data Augmentation (DA) creates new data samples that remain
representative with respect to the original, while introducing diversity into
the training data set. The newly created samples must retain their labels in
the case of classification tasks, or have targets as close as possible to the
original target values in the case of regression problems.

In the case of Image Processing there are geometrical transformations, such as
cropping, flipping, rotation, or translation, among others, that do not modify
the category of the samples. In Natural Language Pocessing, a word can be
replaced by a synonym [4], or some words can be removed to create new
sentences. However, these kind of transformations can not be applied directly
in other scenarios. this is the case of time series data, where those
transformations can not be used due to the particular nature of time series
distributions. For example, a human can easily decide whether a transformed
image or text still retains the original target. However, this is not that
easy with time series data, since geometric transformations could potentially
alter time-domain features [5] without being detected.

Noise addition has been widely used as a DA technique [6, 7, 8]. The method
involves adding random noise to each training sample during training, what
introduces variability to the training sample space, hinding overfitting the
training dataset. This method has been found to be equivalent to Tikhonov
regularization [9], making it intriguing to explore other related techniques
for data enhancement. One approach previously studied consist in training a
model to remove noise previously added to raw samples. This denoising model
must learn the data distribution trough a training process to learn how to
remove noise o reconstruct the raw sample. Since the random noise added can
not be exactly predicted, new samples will be generated hoping to maintain the
time domain features of the raw samples.

This work explores denoising models for data augmentation in time series, with
a focus on both classification and regression tasks. Our contributions can be
summarized as follows:

  * 1.

Introduction of a methodology D3A-TS for applying denoising models to data
augmentation, validated across both classification and regression tasks in
time series (section 3.1).

  * 2.

Investigation into the use of diffusion models for data augmentation in both
classification and regression tasks, a field that lacks prior exploration
(section 4).

  * 3.

Proposal and demonstration of the utilization of a set of meta-attributes to
condition the denoising model, showcasing its efficacy in enhancing denoising
models for data augmentation (section 3.4).

  * 4.

Conducting a comprehensive ablation analysis of the proposed methods to
validate our findings against raw data, noise augmentation, and autoencoders
(section 6.1).

##  2 Previous work

Time series data augmentation methods need to possess the ability to both
produce diverse samples and faithfully replicate the properties of real-world
data [1]. Fundamental DA techniques for time series are based on the
manipulation of time series directly, for instance, deformation, modification,
enlargement or shortening of the raw time series. Some of these algorithms are
adaptations of data augmentation techniques from computer vision, to be
applied in time series, and others are specifically designed for its use in
this field. The main techniques are: flipping , jittering, scaling, rotation
and permutation. These techniques are shown graphically in the figure 1.

Figure 1: A sample of basic data augmentation techniques.

###  2.1 General Deep Learning models for DA

Recent studies in generative Deep Learning (DL) models have made possible to
create synthetic data of high quality. In this area three main architectures
can be found: auto-enconders (AE), generative adversarial networks (GAN), and
the recently successful diffusion probabilistic models (DPM).

####  2.1.1 Generative Adversarial Networks

GAN architecture, introduced in 2014 by Ian Goodfellow [10], is composed by
two networks: the generator and the discriminator. The goal of the generator
network is to generate new samples from a random vector as similar as possible
to the original ones, with the aim of misleading the discriminator. In the
other hand, the discriminator has to predict whether the samples belong to the
original samples or to the generated ones. During the training, the generator
will become better on creating similar samples and the discriminator in
detecting generated samples. In this way, the generator network, once it has
been trained, allows extracting new samples from the learned data
distribution. This type of architecture has been employed to augment data in
various ways. In [11], a GAN is used to tackle issues with extremely
imbalanced datasets by generating samples for the less-represented class.
However, this architecture comes with an important drawback associated to the
training phase [12, 13, 14]. The basic GAN architecture does not require a
reference raw sample to generate synthetic samples. Therefore, to assign a
category to the generated sample, it is necessary to train a conditioned
generator [15, 16, 17, 18].

####  2.1.2 Autoencoders

The AE architecture was introduced by Hinton and the PDP group in 1986 [19] to
tackle the challenge of ”backpropagation without a teacher”, also known as
unsupervised learning. This architecture consists of two networks: the encoder
and the decoder. The encoder’s primary goal is to obtain a low-dimensional
latent representation of the input data, while the decoder’s task is to
reconstruct the original input using this low-dimensional representation. Due
to the limited information in the low dimension representation of the input,
one of the most popular applications of AEs is input denoising [20, 21, 22,
23]. That is, random noise is added to the original sample and this noisy
input is injected in the codificator. The output, given by the decodificator,
will have the noise removed and therefore is expected to be very similar to
the original sample. In addition, once the AE is trained, it is possible to
use the decodificator to sample new data from the latent space.

One problem with the AEs is that the latent space is not well structured, and
therefore, the sampled data could have bad quality. There are other variants
of AE that address this issue [24]. Variational Auto-Encoders (VAE) [25]
mitigates this issue encoding the input using a Gaussian probability
distribution. The latent vectors are generated from a vector of means and
standard deviations, which define the latent space distribution. The
imposition that the latent space must follow a gaussian distribution is
achieved using the Kullback-Leibler Divergence as a regualization term in the
loss function.

VAEs have been used as a data augmentation tool in several works [26, 27, 28].
Most of the applications of VAEs for data augmentation have been carried out
conditioning the decoder to try ensuring the category of the sample generated.
Only a few works have follow a denoising-based DA strategy using VAEs[29].

####  2.1.3 Diffusion Probabilistic Models

DPMs have recently garnered significant attention in the field of Computer
Vision, showcasing remarkable achievements in image generation [30]. These
kind of model consists of two distinct stages: the _forward diffusion_ stage
and the reverse diffusion stage. In the former, the input data undergoes a
gradual transformation through the incremental introduction of Gaussian noise
over multiple steps. In latter, a model is trained reconstruct the original
input data by systematically removing the noise, step by step.

DPMs have been applied to various tasks in time series data, such as time
series forecasting [31, 32], audio signal generation [33, 34], and time series
imputation [35]. However, to the best of our knowledge, there is a notable
absence of studies focused on the utilization of DPMs as a data augmentation
technique for time series.

##  3 Methods

This section introduces the foundational instruments employed in the
methodology for data augmentation in time series, developed in this work. The
section unfolds through four subsections. It commences with an exploration of
Data Augmentation through denoising (section 3.1), leveraging techniques to
feed the models with faithful samples that mirror real-world complexities. In
the subsequent sections, Denoising Autoencoders (section 3.2) and Diffusion
Model (section 3.3), where the specific use of these models for both denoising
and data augmentation approaches is introduced. In the last section (section
3.4), Meta-Attributes for Conditioning Denoising Models, the set of meta-
attributes used to infuse and preserve sample characteristics, augmenting
overall model performance, is defined.

###  3.1 Data Augmentation through denoising

Let $x^{(i)}$ be a time series sample from an input space
$\mathcal{X}=\mathbb{R}^{d}$. A noisy sample, denoted as $\widetilde{x}_{k}$,
is a sample that has noise added through some mapping process:

$\widetilde{x_{k}}=\mathcal{M}(x)=x+\xi_{k}$

Where $\xi_{k}$ is sampled from some noise distribution and $k$ denotes a
sample extraction step. Usually $\xi_{k}$ is sampled from a Gaussian
distribution with mean of 0 and standard deviation of 1;
$\xi_{k}\sim\mathcal{N}(0,I)$. Note that $\xi_{k}$ can be decomposed in the
summation of lower-rate noises $\xi_{k}=\sum_{i=0}^{T}\epsilon_{k,i}$. In this
way, a noisy sample can be seen as a process of progressive noise insertion
$\widetilde{x}_{k,1},\widetilde{x}_{k,2},..,\widetilde{x}_{k,t},..,\widetilde{x}_{k,T}$
where $\widetilde{x}_{k,t}=\sum_{i=0}^{t}\epsilon_{k,i}$.

A denoising model $\mathcal{H}$ can be trained to learn to remove noise
introduced, denoted as $\xi_{k}$. That is, essentially, reconstructing the
sample $x$ from the noisy sample $\widetilde{x}_{k}$, where
$x\approx\mathcal{H}(\widetilde{x}_{k})=\hat{x}_{k}$. Where $\hat{x}_{k}$
represents the reconstructed sample by the model $\mathcal{H}$ and is an
approximation of the source sample $x$. Another approach involves training the
model to remove only part of the noise introduced, i.e.,

$\widetilde{x}_{k,t-1}\approx\mathcal{H}(\hat{x}_{k,t},t)=\hat{x}_{k,t-1}$

When the denoising model $\mathcal{H}$ has learned the data distribution and
the noise added is sufficiently small, the category of the sample $\hat{x}$
should be likely retained.

Another way to enforce the preservation of the category is by conditioning the
denoising model on the category of the sample $x$, expressed as
$\mathcal{H}(\cdot,y)=\hat{x}{\cdot}$, where $y$ denotes the category of the
sample $x$. This approach can be generalized by conditioning the denoising
process on a set of meta-attributes extracted from the sample:
$\mathcal{H}(\cdot,a)=\hat{x}{\cdot}$, where $a=\mathcal{A}(x)$, with
$\mathcal{A}$ representing the process of meta-attribute extraction and $a$
being the meta-attributes vector extracted from $x$.

If the mapping process $\mathcal{M}$ is stochastic, multiple noisy samples can
be generated. This set of noisy samples can then be denoised using the model
$\mathcal{H}$ to produce a set of new samples, which are likely to belong to
the same category than the source sample $x$:

$\\{\mathcal{H}(\mathcal{M}(x,\cdot)),\mathcal{H}(\mathcal{M}(x,\cdot)),..,\mathcal{H}(\mathcal{M}(x,\cdot)),..,\mathcal{H}(\mathcal{M}(x,\cdot))\\}=\\{\hat{x}_{1,\cdot},\hat{x}_{2,\cdot},..,\hat{x}_{k,\cdot},..,\hat{x}_{n,\cdot}\\}$

Therefore, in this way it is possible to extend the data set generating $n$
new samples from each sample $x$. For simplicity, in the rest of the paper the
$k$ parameter will be ommited.

###  3.2 Denoising Autoencoder

A Denoising Autoencoder (DAE) is an autoencoder designed to reconstruct the
original sample $x^{(i)}$ from a noisy observation $\widetilde{x}^{(i)}$.
During the training phase, the model’s parameters need to be optimized by
minimizing the average reconstruction error, which is typically represented by
the mean squared error, as follows:

$\underset{\phi,\theta}{\text{arg
min}}\frac{1}{M}\sum_{i=1}^{M}\mathcal{L}\left(x^{(i)}-f_{\phi}\cdot
g_{\theta}\left(\widetilde{x}^{(i)}\right)\right)$ (1)

In this equation, $\mathcal{L}$ denotes the reconstruction error, $g$ and $f$
represent the encoder and decoder networks, respectively, and $\phi$ and
$\theta$ denote the parameters of the encoder and decoder networks that need
to be learned.

Once the DAE is trained, multiple observations based on $x^{(i)}$ can be
obtained using the DAE:

$\hat{x}^{(i)}=f_{\phi}\cdot g_{\theta}(\widetilde{x})$ (2)

Noted that:

$\mathcal{H}=f\cdot g$

In this work is considered that the demonising process with the autoencoder
can be applied multiples times over the same sample. This is denoted as:

$\begin{split}&\hat{x}_{t-1}^{(i)}=f_{\phi}\cdot
g_{\theta}(\hat{x}_{t}^{(i)});T\geq t\geq 1\end{split}$ (3)

where $t$ denotes the remaining denoising steps.

###  3.3 Diffusion models

In a broader sense, diffusion models adhere to the concept of systematically
eliminating noise over a series of steps. These models are trained to predict
the noise introduced at each step of the process. Consequently, they can be
employed to progressively remove the noise, step by step. The procedure of
introducing noise is referred to as the forward diffusion process, and its
corresponding posterior distribution is denoted as $q$:

$\begin{split}q(x_{1:T}|x)&=q(x_{0})q(x_{1}|x_{0})...q(x_{T}|x_{T-1})\\\
&=q(x_{0})\mathcal{N}(x_{1}|\sqrt{1-\beta_{1}}x_{0},\beta_{1}I)...\mathcal{N}(x_{T}|\sqrt{1-\beta_{T}}x_{T-1},\beta_{T}I)\\\
&=q(x_{0})\prod_{t=1}^{T}\mathcal{N}(x_{t}|\sqrt{1-\beta_{t}}x_{t-1},\beta_{t}I)\end{split}$
(4)

where $I$ is the identity matrix and $\beta_{1},\beta_{2},...,\beta_{T}$
represents an increasing variance schedule with $\beta_{t}\in(0,1)$
controlling the level of noise added at each step. $\mathcal{N}(x|\mu,\sigma)$
denotes the probability density at $x$. It is worth noting that as $T$
approaches infinity, it resembles an isotropic Gaussian distribution.

By using the parametrization trick, it is possible to sample $x_{t}$ at any
arbitrary time step in a closed form, avoiding the need to execute the noise
addition process over T steps. Let $\alpha_{t}=1-\beta_{t}$, and
$\sqrt{\overline{\alpha}_{t}}=\prod_{i=1}^{t}\alpha_{i}$:

$q(x_{t}|x_{0})=\mathcal{N}(x_{t}\sqrt{\overline{\alpha}_{t}}x_{t-1},\overline{\alpha}_{t}I)$
(5)

In order to sample new data, it is necessary to learn the reverse distribution
$q(x_{t-1}|x_{t})$. However, obtaining the closed form of the former posterior
distribution is intractable, as it would require the computation of the
involved data distribution. This issue can be solved by training a
parameterized model $p_{\phi}$ to approximate $q(x_{t-1}|x_{t})$. The model
$p_{\phi}$ can be set to be Gaussian:

$p_{\phi}(x_{t-1}|x_{t})=\mathcal{N}(x_{t-1}|\mu_{\phi}(x_{t},t),\sigma_{\phi}(x_{t},t)I)$
(6)

where $\phi$ represents the parameters of the model to be learned. Therefore,
the model $p_{\phi}$ will be trained to predict both the mean $\mu_{\phi}$ and
variance $\sigma_{\phi}$ for time step $t$.

The training process requires minimizing the negative log-likelihood of the
training data. In [30], authors introduce a loss function that performs better
than the original Evidence Lower Bound (ELBO), denoted as:

$\underset{\phi}{\text{arg
min}}\frac{1}{M}\sum_{i=1}^{M}\mathcal{L}(\epsilon-\epsilon_{\phi}(\sqrt{\overline{\alpha}}_{t}x_{0}+\sqrt{1-\overline{\alpha}}_{t}\epsilon,t))$
(7)

where $\epsilon\sim\mathcal{N}(0,I)$. The previous equation aims to minimize
the noise $\epsilon$, added to the sample $x_{t-1}$, and the error predicted
by $\epsilon_{\phi}$, given a noisy sample, represented as
$\sqrt{\overline{\alpha}}_{t}x_{0}+\sqrt{1-\overline{\alpha}}_{t}\epsilon$ and
the time step $t$. This can be denoted as:

$p_{\phi}(x_{t-1}|x_{t})=\frac{\sqrt{\overline{\alpha}_{t}}x_{0}+\sqrt{1-\overline{\alpha}_{t}}\epsilon-\sqrt{1-\overline{\alpha}_{t}}\epsilon_{\phi}(\sqrt{\overline{\alpha}}_{t}x_{0}+\sqrt{1-\overline{\alpha}}_{t}\epsilon,t)}{\sqrt{\overline{\alpha}_{t}}}$
(8)

###  3.4 Meta-attributes for conditioning denoising models

As mentioned earlier, a denoising model $\mathcal{H}$ can be conditioned on a
set of meta-attributes $a$. In this work, the hypothesis under study is that
this conditioning can enhance the denoising model’s effectiveness in the data
augmentation process by assisting in preserving the source category of the
sample.

For each time series, 15 meta-attributes were extracted. Note that, since the
work is focused on time series data, the meta-attributes has been specifically
selected to work with time series. These are described as follows:

  * 1.

Stability. Squared Pearson correlation coefficient.

  * 2.

Periodicity. The mean Pearson correlation between four equally sized
consecutive segments of the sample.

  * 3.

Oscillation. The absolute of the ratio between the standard deviation and the
mean of the sample.

  * 4.

Complexity. The sum of the absolute values of the Fourier transformation
magnitudes.

  * 5.

Noise. The ratio between the mean and the standard deviation of the sample.

  * 6.

Entropy. The Shannon’s entropy of the sample’s probability distribution.

  * 7.

Variability. The ratio between the standard deviation and the mean of the
sample.

  * 8.

Standard deviation. The standard deviation of the sample

  * 9.

Peculiarity. The sample kurtosis (calculated with the adjusted Fisher-Pearson
standardized moment coefficient G2 [36]

  * 10.

Dynamic range. The absolute value of the difference between the maximum and
minimum values of the sample.

  * 11.

Symmetry. The sample skewness (calculated with the adjusted Fisher-Pearson
standardized moment coefficient G1). [36]

  * 12.

Peaks. The number of peaks that occur at enough width scales and with
sufficiently high Signal-to-Noise-Ratio (SNR) [36]. This feature is computed
from the smoothed sample using the ricker wavelet for widths ranging from 1 to
10.

  * 13.

Slope. The slope of the linear regression of the sample

  * 14.

Min value. The minimum value of the sample

  * 15.

Max value. The maximum value of the sample

##  4 Experiments

This section outlines the methodology employed for conducting experiments.
Initially, a brief description of the datasets used is provided. Subsequently,
the approach adopted to conduct the experiments is explained.

###  4.1 Datasets

Six different datasets have been considered to carry out the experiments on
different type of problmes. Two of these datasets belong to regression tasks,
two to binary classification, and the remaining two to multiclass
classification. Below, each dataset will be briefly described.

####  4.1.1 N-CMAPSS

The N-CMAPSS dataset [37] have been built using the Commercial Modular Aero-
Propulsion System Simulation (CMAPSS), a modeling software developed at NASA.
N-CMAPSS provides full flights trajectories starting with a healthy condition
until a failure occurs. This dataset is gaining popularity in the context of
prognosis and health management (PHM), where works such as [38, 39, 40], among
others, have proposed different solutions to address the problem posed by this
dataset.

The dataset presents seven failure modes associated with either flow
degradation or subcomponent efficiency. The flights are categorized into three
classes based on their duration: Class 1 includes flights lasting 1 to 3
hours, Class 2 covers flights ranging from 3 to 5 hours, and Class 3 comprises
flights lasting over 5 hours. The problem to be addressed with this dataset is
to predict the remaining useful life (RUL) of the system until a failure
occurs. It is assumed that RUL will decrease linearly from the maximum value
(total cycles of the experiment) to 0. The dataset provides 20 input time
series variables, therefore this is a multivariate regression task.

####  4.1.2 PRONOSTIA

PRONOSTIA dataset[41], collected by FEMTO-ST, a French research institute,
using the so called PRONOSTIA platform. The bearings mounted on this platform
were subjected to gradual degradation over time. To expedite this degradation
process, three distinct operating conditions were considered, which
respectively involved rotating speeds of 1800 rpm with a 4000 N payload
weight, 1650 rpm with 4200 N, and 1500 rpm with 5000 N.

Two sensors, placed along the x-axis and y-axis of the bearing, were employed
for data acquisition. Data were recorded every 10 seconds during 0.1 seconds,
with a frequency of 25.6 kHz, therefore, resulting in each time series
comprising 2560 data points. The experiment concluded when the vibration
amplitude exceeded the threshold of 20 g. Similar to the previous dataset, it
is assumed that the remaining useful life (RUL) ofthe system decreases
linearly from the maximum value (total time of the experiment) to 0. However,
for this study, the RUL is normalized to be between 0 and 1.

####  4.1.3 ECG5000

The ECG5000 dataset, sourced from the University of California, Riverside,
serves as a prominent benchmark for time series classification [42]. It stands
as one of the most widely used public datasets for time series classification.
The dataset comprises a 20-hour long ECG (Electrocardiogram) data stream, and
encompasses 5 distinct categories. Each data instance is a sequence with a
length of 140 data points.

The dataset consists of 500 samples for training and 4500 for testing.

####  4.1.4 Human Activity

The Human Activity (ha dataset (HR) [43] is focused on Human Activity
Recognition (HAR). The experiments, carried out to build the dataset, involved
a group of 30 volunteers, each performing six activities (walking, walking
upstairs, walking downstairs, sitting, standing and laying) while wearing a
smartphone (Samsung Galaxy S II) at the waist. The accelerometer signal from
the smartphone was captured, preprocessed to eliminate noise, and divided into
128-point time series, each associated with a specific action. The training
set comprises 7352 samples, while the test set consists of 2947 samples.

####  4.1.5 Wine

The Wine dataset [44] consists of spectrographs from two types of wine:
cabernet sauvignon and shiraz. These signals were obtained using FTIR
spectroscopy with attenuated total reflectance (ATR) sampling.

The dataset contains 57 samples for training and 54 for testing, with each
sample comprising 234 data points.

####  4.1.6 Shares

Finally, the Shares dataset [44] is designed for predicting whether a share
price will experience a notable increase after the quarterly announcement of
earnings, based on the price movement in the preceding 60 days. Each data
point represents the percentage change in the close price with respect to the
day before over a 60-day period. The target class is assigned a value of 1 if
the company stock increased by more than 5 percent after the report release,
and 0 otherwise. The dataset comprises 1931 samples, with 1326 instances
classified as class 0 and 605 as class 1.

Table 1 summarizes the main features of the selected datasets. These datasets
have been sourced from different areas, involve diverse tasks, and vary in
dimensions, to cover a wide variety of experiments. Figure 2 showcase a few
samples from each dataset, highlighting the diversity of shapes explored in
this paper.

Dataset | Task | Area | I.D. | Length | O.R. | Train / Test size
---|---|---|---|---|---|---
N-CMAPSS | R | I | 20 | >3e5 | [0-100] | 195e5 / 840e5
PRONOSTIA | R | I | 2 | 2560 | [0-1] | 10240 / 7680
SHARES | B | F | 1 | 60 | {0,1} | 965 / 965
WINE | B | P | 1 | 234 | {0,1} | 57 / 54
ECG5000 | M (5) | H | 1 | 140 | {0,1,2,3,4} | 500 / 4500
HR | M (6) | CS | 1 | 128 | {0,1,2,3,4,5} | 7352 / 2947
Table 1: Summary of the main features for the six datasets used in this work.
I.D.: Input Dimension, O.R.: Output Range, R: Regression, B: Binary, M (x):
Multi class with x categories, I: Industry, F: Financial, P: Physics, H:
Health, CS: Computer Science. Figure 2: Samples from the six datasets used in
this work. It can be observed that there exists a lot of diversity in the
shapes of the samples.

###  4.2 Experiment settings

Before executing any experiment, it is necessary to generate the meta-
attributes $a^{(i)}$ associated with each sample $x^{(i)}$. In this study, the
samples have been truncated to have 128 data points (64 data points in the
case of the shares dataset). For each segment of 32 data points, 15 meta-
attributes are extracted. As a result, the vector $a^{(i)}$ will have a size
of 60 elements (30 in the case of the shares dataset). It’s important to note
that this process is computationally intensive.

To address this computational complexity, the meta-atributes vectors $a^{(i)}$
are approximated using a single fully connected neural network
$\mathcal{A}_{\psi}$. The network $\mathcal{A}_{\psi}$ comprises three hidden
layers with 32, 64, and 128 neurons, where the hyperbolic tangent (tanh)
function is utilized as the activation function.

The model $\mathcal{A}_{\psi}$ is used while training the denoising model
$\mathcal{H}_{\phi}$ to learn to remove the noise from a noisy sample. The
model $\mathcal{H}_{\phi}$ is composed of 4 downsampling blocks and 4
upsampling blocks. After downsampling, two additional blocks are applied (see
Figure 3). Each block applies one-dimensional operations to the input.
Subsequently, the time step embedding and the attribute embedding are
integrated with the feature map using sum and concatenation operations. Then,
two residual blocks are applied. These embeddings are created using a fully
connected layer and the corresponding reshaping to match the dimensions of the
former one-dimensional convolutional layer.

Figure 3: Left: the upsampling and downsampling block used to create the
denoising model architectures for both autoencoders and diffusion models.
Right: the denoising model architecture composed of four downsampling steps,
two additional blocks, and four upsampling steps.

Finally, the model $\mathcal{H}_{\phi}$ is used to augment data during the
training of the model to solve the classification or regression task. It is
worth noting that this final model is exclusively trained using synthetic
data. These final models are trained using 1, 2, and 3 denoising steps, along
with various noise rates introduced to the samples before applying denoising.
Both the number of denoising steps and the noise rate are considered hyper-
parameters of the training process. The stopping criterion for the training
procedure is early stopping, which halts the training of a model once its
performance on a validation set has not improved after a predefined number of
training steps. This process is summarised in the algorithm 1 and depicted in
figure 4.

Figure 4: The graph illustrates the training process. $\mathcal{A}_{\psi}$
represents the network used to predict the meta-attributes vector $a$ from the
training raw data. $\mathcal{M}$ denotes the process that introduces normal
noise to the raw samples, while $\mathcal{H}_{\phi}$ represents the denoising
network responsible for generating the synthetic samples. The model
$\mathcal{F}$ is then trained using these synthetic samples and validated
against the raw samples from the test set. Algorithm 1 Training algorithm

Set $n$ as the number of denoising steps.

Set $r$ as the noise ratio.

$X$ is a set of samples.

$Y$ is the set of target values $y^{(i)}$ for each $x^{(i)}\in X$

$A$ is a set of meta-attributes extracted from each sample $x^{(i)}\in X$

$\mathcal{M}(x,r)$ is a mapping process that introduces noise with the form
$r\mathcal{N}(0,1)$

Train a model $\mathcal{A}_{\psi}$ to predict $a^{(i)}\in A$ from $x^{(i)}\in
X$

Train a denoising model
$\mathcal{H}_{\phi}(x_{t-1}^{(i)},\mathcal{A}_{\psi}(x^{(i)}))$ to predict
$x_{t}^{(i)}$

Train a model
$\mathcal{F}_{\theta}(\mathcal{H}_{\phi}(\mathcal{M}(x^{(i)},r),\mathcal{A}_{\psi}(x^{(i)}))$
to predict $y^{(i)}\in Y$

In addition to the final model trained with denoised (synthetic) data, final
models using the raw and noisy samples ,respectively, have also been trained
to be compared with such a model. Different experiments were carried out in
which, DPM (Denoising Process Models) and AE (Autoencoder) models are used as
denoising models.

Two different network architectures were tested with each dataset. The first
one is a multi-scale convolutional neural network [45] with two multi-scale
blocks and two additional convolutional blocks, each composed of two stacked
convolutional layers and max-pooling operations. Following the convolutional
layers, two fully connected layers are added before the output layer. The
second architecture is a recurrent network composed of two stacked
bidirectional LSTM layers and two fully connected layers. Mean squared error
was used as the loss function for regression, and softmax loss was used for
classification.

The complete set of experiment settings along with the results obtained in
each one and the source code used to execute all these experiments can be
checked in the GitHub repository https://github.com/DatrikIntelligence/D3A-TS-
Denoising-Driven-Data-Augmentation-in-Time-Series.

##  5 Bayesian test

Due to the well-documented issues with the null hypothesis significance test
[46], this work adopts a Bayesian approach to confront the results obtained.
Specifically, the Bayesian signed-rank test [47] is used, which serves as a
Bayesian adaptation of the Wilcoxon signed rank test. This test takes as input
the paired results of two models, A and B, and provides three probabilities:
the likelihood that model A is better than B, the likelihood that model B is
better than A, and the probability that there are no significant differences
(rope).

The ratio between model A and model B is adopted as the paired input, computed
as:

$\mathcal{R}(a,b)=\frac{a-b}{0.5(a+b)}$ (9)

Here, $a$ and $b$ represent the losses obtained by models A and B,
respectively. $R$ will take a negative value when model A outperforms model B
(the loss error of A is lower than the loss error of B), and a positive value
in the opposite case. It is considered that a value of $R\in[-0.05,0.05]$
means that there are not significant differences (rope).

##  6 Main results

A considerable number of experiments have been executed, involving six
datasets, 2 final networks, 3 data augmentation techniques, and around 10
different noise rates for the 2 denoising models, each with 3 repetitions per
trained model, which makes a total of 1800 experiments. The results highlight
that using the diffusion model for data augmentation in time series problems,
for both classification and regression, is entirely plausible. Additionally,
the results demonstrate that the use of meta-attributes improves the behavior
of the denoising models by preserving the sample category, as hypothesized in
Section 3.4.

Figure 5 displays examples of raw samples, along with their noisy version and
the results obtained after applying denoising with autoencoders and diffusion
models. The autoencoders and diffusion models have been trained both in an
unconditional and conditional way.

Figure 5: Samples from the six datasets used in this work, along with examples
showing the noisy samples after noise addition and the reconstructions made by
the autoencoders and diffusion models, both with unconditional and conditional
denoising.

It is important to emphasize that the primary objective of this work is not to
achieve the model with the lowest loss for each datasets. The focus is
primarily on assessing whether the denoising process can lead to better
generalization of a model, irrespective of the architecture, hyperparameters,
and the data involved. Consequently, the hyperparameters of the regression or
classification model have not been extensively optimized. For each dataset,
different experiments have been carried out to evaluate the proposed methods,
that is, denoising data augmentation based on AE and DPM, with and without
meta-attributes conditioning. Additionally, models have been trained with
noise-based data augmentation. For each of these experiments, another has been
performed using raw data, to assess whether denoising data augmentation based
models improve performance. Each experiment has been repeated three times to
compute the mean loss and compare the results among the different approaches.
In the case of data augmentation methods (noise, AE, and DPM), different noise
rates $\beta_{t}$ have been tested. Finally, tor denoising methods, 1, 2, and
3 denoising steps $t$ have been evaluated.

Table 2 presents the results achieved using denoising conditioned data
augmentation, which are better than the average performance of the models
trained using raw data. As observed, in all instances, it is possible to
identify hyperparameters that surpass the models trained solely on raw data.
Additionally, in most cases, a specific set of hyperparameters demonstrates
that the DPM data augmentation outperforms the AE data augmentation. Figure 6
shows the results of the Bayesian signed-rank test comparing DPM and AE
conditioned data augmentation with the models trained using raw data. The
Bayesian signed-rank test confirms that finding good hyperparameters in DPM
has a high probability, while using AE, the probability is very low.

Dataset | Net | Raw | AE | DPM
---|---|---|---|---
ecg5k | rnn | 0.5669 ± 0.1651 | 0.6932 ± 0.4601 [1] |  0.3235 ± 0.0085 [1]
ecg5k | cnn | 0.6631 ± 0.2387 | 0.5204 ± 0.0325 [3] |  0.4272 ± 0.0205 [1]
HA | rnn | 1.2370 ± 0.2486 | 1.1023 ± 0.0016 [1] |  1.0897 ± 0.0011 [2]
HA | cnn | 1.2821 ± 0.1238 |  1.2461 ± 0.0221 [2] | 1.2758 ± 0.0120 [2]
ncmapss | rnn | 252.9270 ± 13.9745 |  242.1542 ± 0.0000 [2] | 246.7824 ± 16.9870 [2]
ncmapss | cnn | 459.5337 ± 165.5178 | 324.1614 ± 16.9442 [2] |  266.0854 ± 13.5774 [1]
pronostia | rnn | 0.0720 ± 0.0063 | 0.0700 ± 0.0013 [1] |  0.0662 ± 0.0007 [1]
pronostia | cnn | 0.0614 ± 0.0044 |  0.0487 ± 0.0030 [1] | 0.0522 ± 0.0029 [3]
shares | rnn | 0.3480 ± 0.0266 | 0.3435 ± 0.0142 [3] |  0.2947 ± 0.0495 [1]
shares | cnn | 1.2505 ± 0.8296 | 0.3113 ± 0.0157 [3] |  0.2153 ± 0.0175 [3]
wine | rnn | 0.4097 ± 0.0203 | 0.3850 ± 0.0074 [3] |  0.3432 ± 0.0043 [1]
wine | cnn | 1.3288 ± 0.5098 |  0.3298 ± 0.0269 [2] | 0.4529 ± 0.0000 [2]
Table 2: The table compares the best mean results obtained by applying
denoising conditioned data augmentation with the mean performance of models
trained using raw data. The number in brackets refers to the number of
denoising steps applied. Figure 6: Comparative performance of a network
trained with denoising data augmentation and the same network trained with raw
data.

###  6.1 Ablation study

This section explores the effectiveness of denoising-based data augmentation,
with and without meta-attribute conditioning, compared to data augmentation
involving only noise introduction.

####  6.1.1 Denoising process vs Noise augmentation

Figure 7 (left) displays the results of the Bayesian signed-rank test
comparing conditioned noise-based data augmentation against models trained
solely with raw data. As observed, noise data augmentation exhibits a low
probability of generating a better generalized model when compared to models
trained solely with raw data. Figure 7 (center and right) depicts the
likelihood for the denoising-based data augmentation approaches (using DPM and
autoencoders respectively) to perform better than the noise-based approach.
Results distinctly demonstrates the superiority of DPM-based denoising models
as data augmentation tool.

Figure 7: The chart on the left displays the comparative performance between
conditioned noise-based data augmentation and the model trained solely with
raw data. The center and right charts exhibit the performance obtained using
denoising data augmentation based on DPM and autoencoders compared with that
obtained using noise-based data augmentation.

####  6.1.2 Meta-attribute conditioning

Figure 8 shows the training progress of the denoising models. To compute the
confidence intervals and compare the contribution of the meta-attributes, the
validation loss has been normalized per dataset. Results show that meta-
attribute conditioning considerably improves the validation loss and reduces
uncertainty.

Figure 8: Normalized validation loss during the denoising model training
process. The graph shows the confidence interval computed using the validation
loss obtained for each dataset. Both graphs demonstrate that meta-attribute
conditioning significantly contributes to the denoising learning process.

Figure 9 sheds light on the influence of meta-attribute conditioning when
employing denoising-based data augmentation approaches. According to the
results, it appears that the use of meta-attribute conditioning does not have
a clear impact when combined with the AE models. However, when utilized
alongside DPM models, it notably enhances the data augmentation process.

The lack of effectiveness in meta-attribute conditioning, when combined with
the AE approach, might contribute to the superior performance of the DPM
denoising approach. This hypothesis is supported when comparing non-
conditioned approaches with raw data, as illustrated in Figure 10. The removal
of conditioning in the DPM drastically reduces the efficiency of the data
augmentation process. However, the results of the AE remain very similar to
those shown in Figure 6.

Figure 9: The chart depicts the influence of meta-attribute conditioning in
the DPM and AE denoising data augmentation approaches. On the left, it is
noticeable that conditioning substantialy impacts DPM approach. Conversely,
the chart on the right demonstrates no impact of meta-attribute conditioning
when AE are employed. Figure 10: These charts depict the impact of
unconditional denoising models compared to models trained solely with raw
data. The comparison with Figure 6 highlights the influence of the
conditioning, especially in DPM.

###  6.2 Number of denoising steps $T$ and noise rates $\beta_{T}$

Figure 11 depicts the search space of the hyperparameters $T$ (number of
denoising steps) and $\beta_{T}$ (noise rate added to the original sample).
The charts were created by fitting a Gaussian model with the results obtained
during the experimentation, and creating an interpolation between the extreme
values using the predictions of this Gaussian model. The analysis of this
graph demonstrates that these hyperparameters have to be selected carefully,
since the hyperparameter search space changes with each dataset and model
architecture used. In general, the charts show that there are extensive areas
where good configurations of these hyperparameters can be found.

Figure 11: Comparison of the prediction performance using different denoising
rates and denoising steps in denoising models. The y-axis of the charts
corresponds to the number of denoising steps, and the x-axis corresponds to
the noise rates added to the samples. Blue regions denote better performance
than red regions.

##  7 Conclusions and Future work

This work has studied the application of denoising models for data
augmentation in time series for classification and regression tasks.
Additionally, a set of meta-attributes extracted from the raw data has been
proposed to condition the network with the idea of enforcing the preservation
of the sample category. Data expansion has been evaluated using single noise
addition, denoising autoencoders (DAEs), and diffusion probabilistic models
(DPMs) not previously studied in this area.

The study results have exhibited promising outcomes, demonstrating the
potential of denoising models for effective data augmentation in diverse time
series applications. Furthermore, the inclusion of meta-attributes has proven
to be informative, enhancing the denoising process and providing a more
effective data augmentation tool.

Beyond the promising results obtained, the proposed methodologies suffer some
limitations. As a common issue previously demonstrated in autoencoders [48],
the introduced bias in the input will result in bias in the generated output.
Thus, the denoising steps and noise rates need to be chosen carefully for the
model to be applied successfully. Another important issue is the time
requirements for denoising, which can make the training process slower. These
issues will be the aim of a future work.

The proposed approaches have a excellent potential to be useful as data
augmentation tools, but further work is required. The mentioned techniques
introduce some hyperparameters (noise level and denoising steps) that either
need optimization or exploration in some way. For instance, to avoid manual
design, it might be interesting to learn how to apply an augmentation policy
where different noises and denoising parameters are applied during the
training process, similar to other works such as [49] or [50]. With the recent
rise of large language models (LLM), it could be interesting to study whether
this type of model, using the proper prompts, could help supporting the former
task.

The meta-attributes have demonstrated to be sufficiently informative to
enhance the denoising process. Additionally, they provide more descriptive
information about the features of the sample. Therefore, a denoising model
conditioned with the meta-attributes could be utilized to generate a
neighborhood of a sample. This neighborhood could be employed in explainable
artificial intelligence (XAI) methods such as LIME or SHAPLEY to explain the
local prediction based on the meta-attributes variations. We believe that this
idea could be useful in contexts like process optimization, predictive
maintenance, or root cause analysis, where the application of XAI methods is
more rare [51, 52].

In a similar fashion, the process of generating new samples while preserving
the statistical information supported by the meta-attributes could be employed
in Differential Privacy (DP), as recent works explore in this direction [53,
54].

## Acknowledgments

This work has been supported by Proyecto PID2019-109152GB-I00 financiado por
MCIN/ AEI /10.13039/501100011033 (Agencia Estatal de Investigación), Spain and
by the Ministry of Science and Education of Spain through the national program
“Ayudas para contratos para la formación de investigadores en empresas
(DIN2019-010887 / AEI / 10.13039/50110001103)”, of State Programme of Science
Research and Innovations 2017-2020.

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# Convex bodies with equipotential circles

Iván González-García1, Jesús Jerónimo-Castro2
Valentín Jiménez-Desantiago3, and Efrén Morales-Amaya4
1,2Facultad de Ingeniería
Universidad Autónoma de Querétaro, México
3Instituto de Matemáticas,
Universidad Nacional Autónoma de México, México
4Facultad de Matemáticas-Acapulco,
Universidad Autónoma de Guerrero, México

###### Abstract

Given a convex body $K\subset\mathbb{R}^{2}$ we say that a circle
$\Omega\subset\text{int}K$ is an _equipotential circle_ if every tangent line
of $\Omega$ cuts a chord $AB$ in $K$ such that for the contact point
$P=\Omega\cap AB$ it holds that $|AP|\cdot|PB|=\lambda$, for a suitable
constant number $\lambda$. The main result in this article is the following:
_Let $K\subset\mathbb{R}^{2}$ be a convex body which has an equipotential
circle $\mathcal{B}$ with centre $O$ in its interior. Then $K$ has centre of
symmetry at $O$, moreover, if none chord of $K$ which is tangent to
$\mathcal{B}$ subtends an angle $\pi/2$ from $O$, then $K$ is a disc._ We also
derive some results which characterizes the ellipsoid and the sphere in
$\mathbb{R}^{3}$ and introduce also the concept of equireciprocal disc.

##  1 Introduction

Let $\Gamma$ be a closed convex curve in the plane, i.e., the boundary of a
compact and convex figure in the plane, and let $P$ be a point in its
interior. We say that $P$ is an _equichordal point_ if every chord of $K$
through $P$ have the same length. There is a very famous problem due to W.
Blaschke, W. Rothe, and R. Weitzenböck [3], which asks if there exists a
convex body with two equichordal points. There is a long story, full of many
false proofs about the no existence of such a body. In [15] M. Rychlik finally
gave a complete proof about the no existence of a body with two equichordal
points. However, there are many convex bodies, different from the disc, which
have exactly one equichordal point. One example of these bodies is the Limaçon
of Pascal shown in Fig. 1, moreover, this figure has also the property that
the locus of midpoints of the chords through the equichordal point is a
circle.

Figure 1: The Limaçon of Pascal has an equichordal point

Another interesting kind of points is the following: we say that $P$ is an
_equipotential point_ if for every chord $AB$ of $K$, through $P$, the product
of the lengths $|AP|\cdot|PB|$ is equal to a given constant value. Clearly,
every point in the plane is an equipotential point if $\Gamma$ is a circle.
Indeed, if with respect to a given curve $\Gamma$, every point $P$ in the
plane is equipotential, then $\Gamma$ is a circle. The proof of this fact is
easy and we let it to the interested reader. A more interesting question is:
how many equipotential points with respect to a curve $\Gamma$, ensure us that
it is a circle?

The answer to the above question is 2, as was proved by K. Yanagihara in [20].
Some years later, J. Rosenbaum also proved that two equipotential points are
enough [14]. That only one equipotential point is not enough is shown by the
curve already given by Yanagihara [20]. Let $ABCDEF$ be a regular hexagon with
centre $O$; consider the circumscribed circles to the triangles $\triangle
OBC$, $\triangle ODE$, and $\triangle OFA$. The curve formed by the segments
$AB$, $CD$, $EF$, and the arcs $\widehat{BC}$, $\widehat{DE}$, and
$\widehat{FA}$, has $O$ as an equipotential point.

Figure 2: A curve with an equipotential point $O$

We would like to extend the notion of equipotential and equichordal points to
fatter sets of points.

######  Definition 1.

Given a convex body $K\subset\mathbb{R}^{2}$ we say that a circle
$\Omega\subset\emph{int}K$ is an _equipotential circle_ (_equichordal circle_)
if every tangent line of $\Omega$ cuts a chord $AB$ in $K$ such that for the
contact point $P=\Omega\cap AB$ it holds that $|AP|\cdot|PB|=\lambda$ (it
holds that $|AB|=\lambda$), for a suitable constant number $\lambda$.

In [2], J. A. Barker and D. G. Larman proved that the only convex body in the
plane which possesses an equichordal circle is the disc. The question now is,
what convex bodies, besides the discs, possesses an equipotential circle?

By the Example 37 of Chapter XII in the book by C. Smith, we know that the
envelope of the chords in an ellipse, which are seen under an angle of $\pi/2$
from the centre of the ellipse, is a circle. Indeed, this is the unique
equipotential circle for the ellipse. For a given ellipse $K$, this
equipotential circle is constructed as follows: suppose the centre of $K$ is
$O$, the minor and mayor axes $AC$ and $BD$, respectively. Let $\Omega$ be the
circle inscribed in the rombus $ABCD$ and let $P$ be the point of tangency
between $AB$ and $\Omega$. Since $\measuredangle AOB=\pi/2$, we have that
$|AP|\cdot|PB|=|PO|^{2}=r^{2},$ where $r$ is the radius of $\Omega$. We
reformulate our question in a more precise way: what convex bodies, besides
discs and ellipses, possesses an equipotential circle?

Figure 3: Ellipses have equipotential circles

We solved almost completely this question in Theorem 1 and use the result to
give some characterizations of the circle in $\mathbb{R}^{2}$, the sphere and
the ellipsoid in $\mathbb{R}^{3}$. We also deal with one more kind of circles:
_equireciprocal circles_.

##  2 Convex bodies with equipotential circles

The main result in this section is the following.

######  Theorem 1.

Let $K\subset\mathbb{R}^{2}$ be a convex body which has an equipotential
circle $\mathcal{B}$ with centre $O$ in its interior. Then $K$ has centre of
symmetry at $O$, moreover, if none chord of $K$ which is tangent to
$\mathcal{B}$ subtends an angle $\pi/2$ from $O$, then $K$ is a disc.

Before we give the proof of Theorem 1, we will give some useful lemmas. Let
$AB$ be any chord of $K$ which is tangent to $\mathcal{B}$ at a point $P$ and
such that $\mathcal{B}$ is at the left of the directed segment
$\overrightarrow{AB}$. Let $F$ be the map that takes $A$ to $B$; clearly, $F$
is an orientation preserving homeomorphism of $\partial K.$

######  Lemma 1.

The map $F$ of the curve $\partial K$ has an invariant measure and hence is
conjugated to a circle rotation.

_Proof._ Let $ds$ be the arc length element and let $\rho(A)=|AP|$ be the
length of the tangent segment from $A$ to the circle. Consider the measure
$d\mu=\frac{ds}{\rho}$, the measure of any Borel set $M\subset\partial K$ is
given by

$\mu(M)=\int\limits_{M}\frac{ds}{\rho}.$

The measure $\mu$ is an $F$-invariant measure, i.e.,

$\mu(M)=\mu(F(M)).$ (1)

Indeed, let $A^{\prime}$ be a point infinitesimally close to $A$ and
$B^{\prime}=F(A^{\prime}).$ Then, by the condition that the product
$|AP|\cdot|PB|$ is a constant independent of the initial point $A$, the
infinitesimal triangles $\triangle AA^{\prime}P$ and $\triangle BB^{\prime}P$
are similar, and hence
$\frac{|AA^{\prime}|}{|AP|}=\frac{|BB^{\prime}|}{|B^{\prime}P|}$, that is,

$\frac{dA}{\rho(A)}=\frac{dB}{\rho(B)},$ (2)

(we use the fact that the two tangent segments to a circle from any point are
equal). Integration of (2) gives (1). Now we apply Denjoy’s theorem (see for
instance Theorem 12.3 in [5]) and conclude that $F$ is conjugate to a circle
rotation. $\Box$

Figure 4: Triangles $\triangle AA^{\prime}P$ and $\triangle BB^{\prime}P$ are
similar

######  Lemma 2.

Under the application of the map $F$ all orbits are periodic with the same
period or all are dense.

_Proof._ This is a direct consequence of Lemma 1. $\Box$

We are ready now to prove the main result in this section.

_Proof of Theorem 1._ Let $A_{0}A_{1}$ be any chord of $K$ tangent to
$\mathcal{B}$. Let $A_{2}$ be the point in $\partial K$ such that $A_{1}A_{2}$
is also tangent to $\mathcal{B}$, an so on. Denote by $\alpha=\measuredangle
A_{0}OA_{1}$, if $\frac{\alpha}{\pi}$ is an irrational number, then the orbit
of the point $A_{0}$ is dense in the boundary of $K$. Denote by $P_{i}$ the
point of tangency between the segment $A_{i}A_{i+1}$ and $\mathcal{B}$, for
every $i=0,1,2,\ldots$. Since $|P_{0}A_{1}|=|A_{1}P_{1}|$ and
$|A_{0}P_{0}|\cdot|P_{0}A_{1}|=|A_{1}P_{1}|\cdot|P_{1}A_{2}|$, we have that
$|A_{0}A_{1}|=|A_{1}A_{2}|$. Continuing in this way we see that all the chords
$A_{i}A_{i+1}$ have the same length. By a theorem due to J. Barker and D.
Larman given in [2] we have that $K$ is a disc. Suppose now that
$\frac{\alpha}{\pi}$ is a rational number, then the orbit of the point $A_{0}$
is periodic with period $n$, for some natural and fixed number $n\geq 3$,
independently of the initial point $A_{0}$. Moreover, since the length of the
chord $A_{0}A_{1}$ depends continuously on the point $A_{0}$, and every orbit
has period $n$, we have that all the chords of $K$ and tangent to
$\mathcal{B}$ are seen under the same angle $\alpha$. Now we will prove the
following:

Claim 1. The length of all the chords of $K$ tangent to $\mathcal{B}$ is a
constant.

_Proof._ Suppose the point of contact between $A_{0}A_{1}$ and $\mathcal{B}$
is $P$ and the angle $\measuredangle A_{1}OP=\theta$. Without loss of
generality we may suppose that the radius of $\mathcal{B}$ is 1 (see Fig. 5).

Figure 5: It holds that
$|A_{1}P|\cdot|PA_{0}|=\tan\theta\cdot\tan(\alpha-\theta)$

We have that

$|A_{1}P|\cdot|PA_{0}|=\tan\theta\cdot\tan(\alpha-\theta)=\lambda,$

for some positive constant $\lambda$. We know that given the admissible value
of $\lambda$, the following is true:

Claim 2. If $\alpha\neq\frac{\pi}{2}$ then there are only two values of
$\theta$ that solves the equation
$\tan\theta\cdot\tan(\alpha-\theta)=\lambda$, indeed, if one of the solutions
is $\theta_{0}$ the other one is $\alpha-\theta_{0}$.

_Proof._ By the trigonometric identity for tangent of sum of angles we have
that

$f(\theta):=\tan\theta\cdot\tan(\alpha-\theta)=1-\frac{\tan\theta+\tan(\alpha-\theta)}{\tan\alpha}.$

Deriving the function $f$ we obtain
$f^{\prime}(\theta)=-\frac{1}{\tan\alpha}\left(\sec^{2}\theta-\sec^{2}(\alpha-\theta)\right).$
If $\alpha=\pi/2$ then $\lambda$ must be equal to 1, $f^{\prime}(\theta)=0$
and so every $\theta\in(0,\pi/4]$ gives an admissible solution. Now we
consider the case $\alpha\neq\pi/2.$ Since the function $\sec^{2}(\theta)$ is
an increasing function in the interval $[0,\pi/2)$, we have that
$\sec^{2}\theta-\sec^{2}(\alpha-\theta)<0$, for every $\theta\in(0,\alpha/2],$
this implies that $f^{\prime}(\theta)>0$, for every $\theta\in(0,\alpha/2].$
If follows that the function $f(\theta)$ is strictly increasing in the
interval $\theta\in(0,\alpha/2],$ hence for every admissible value of
$\lambda$ there are at most two values of $\theta\in(0,\alpha)$ which solves
the equation. Indeed, there are exactly two values if $\alpha\neq\pi/2$,
namely, $\theta$ and $\alpha-\theta.$ $\Box$

Claim 2 implies that there is only one value for the length of the chord
$A_{0}A_{1}$. This ends the proof of Claim 1. $\Box$

Continuing with the proof of Theorem 1, we have that in the case when $A_{0}$
has a periodic orbit, all the chords of $K$ tangent to $\mathcal{B}$ have the
same length. We use Barker-Larman’s theorem and conclude that $K$ is a disc.

Finally, if $\alpha=\frac{\pi}{2}$ then the orbit of $A_{0}$, for every
$A_{0}\in\partial K$, has period equal to 4. This means that $O$ is the
midpoint of $A_{0}A_{2}$, for every $A_{0}\in\partial K,$ i.e., $K$ has centre
of symmetry at $O$. $\Box$

######  Remark 1.

If $\alpha=\frac{\pi}{2}$ then $|A_{1}P|\cdot|PA_{0}|=|PO|^{2}$, but the same
is true if $Q$ is any point in the arc of $\mathcal{B}$ contained in the angle
$\measuredangle A_{0}OA_{1}$, i.e., we also have that
$|B_{1}Q|\cdot|QB_{0}|=|QO|^{2}=|PO|^{2}$ (see Fig. 6). This implies that the
chords of $K$ tangent to $\mathcal{B}$ can have an infinite number of values
for its length. Under this condition is not possible to say that $K$ is disc.

Figure 6: It holds that $|A_{1}P|\cdot|PA_{0}|=|B_{1}Q|\cdot|QB_{0}|$,
however, $A_{0}A_{1}\neq B_{0}B_{1}$

######  Example 1.

Besides the example of the ellipse given in the introduction, we have the
following: consider the arc of the parabola with focus in
$(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})$ and directrix $y=-x+2\sqrt{2}$, in
the first quadrant of the Cartesian coordinate system. From every point $A$ in
this arc, consider the tangent line to $\mathcal{B}$ and let $B$ be the point
in this line such that $|BP|\cdot|PA|=1.$ The locus of the point $B$ is a
convex curve in the second quadrant and shares the tangent line with the
parabola at the point $(0,\sqrt{2})$. Now, consider the symmetric image of the
union of these two arcs, with respect to $O$. In this way we obtain the
boundary of a convex body $K$ which has $\mathcal{B}$ as its equipotential
circle (see Fig. 7).

Figure 7: The body $K$ has an equipotential circle

We will explain why this body $K$ is convex. For the sake of clarity we rotate
the parabola an angle $3\pi/4$ in the clockwise sense. Let $\Gamma$ be the
parabola whose equation in Cartesian coordinates is
$y=\frac{x^{2}}{2}-\frac{3}{2},$ and let $\gamma$ be the locus of the point
$B$. The arc of the parabola in the first quadrant in Fig. 7 corresponds to
the arc $\widehat{CD}$ in Fig. 8. By the similarity of the triangles
$\triangle OAP$ and $\triangle OBP$ we have that

$\frac{|OA|}{|OB|}=\frac{|AP|}{|OP|}=|AP|,$

hence $|OB|=\frac{|OA|}{|AP|}.$

Figure 8: It holds that $|OB|=\frac{|OA|}{|AP|}$

Let $\gamma^{\prime}$ be the image of $\gamma$ under a rotation of an angle
$\pi/2$ about $O$ in the clockwise sense. Let $B^{\prime}$ be the image of the
point $B$ under this rotation. We have that $|OB^{\prime}|=\frac{|OA|}{|AP|}.$
The length of the segment $AP$ is
$|AP|=\sqrt{-\frac{x^{2}}{2}+\frac{x^{4}}{4}+\frac{5}{4}}.$ If we parametrize
$\Gamma$ and $\gamma^{\prime}$ in polar coordinates by $(\theta,r(\theta))$
and $(\theta,r_{1}(\theta))$, respectively, we have that

$r_{1}(\theta)=\dfrac{r(\theta)}{\sqrt{-\dfrac{r^{2}(\theta)\cos^{2}\theta}{2}+\dfrac{r^{4}(\theta)\cos^{4}\theta}{4}+\dfrac{5}{4}}},$

where $r(\theta)=\frac{\sin\theta+\sqrt{1+2\cos^{2}\theta}}{\cos^{2}\theta}.$
Since $r^{2}(\theta)\cos^{2}\theta-2r(\theta)\sin\theta-3=0,$ we obtain

$r_{1}(\theta)=\dfrac{r(\theta)}{\sqrt{r^{2}(\theta)\sin^{2}\theta+2r(\theta)\sin\theta+2}}.$

We recall that the curvature in polar coordinates is calculated by

$\kappa(\theta)=\dfrac{\rho^{2}(\theta)-\rho(\theta)\rho^{\prime\prime}(\theta)+2(\rho^{\prime}(\theta))^{2}}{((\rho^{\prime}(\theta))^{2}+\rho^{2}(\theta))^{\frac{3}{2}}},$

so if we want to know that $\gamma^{\prime}$ is a convex curve, we just need
to see
that$r_{1}^{2}(\theta)-r_{1}(\theta)r_{1}^{\prime\prime}(\theta)+2(r^{\prime}_{1}(\theta))^{2}>0$,
for every $\theta\in[0,2\pi].$ This is indeed the case, we omit the simple but
tedious computations, hence we have that $\gamma^{\prime}$ is a convex curve,
i.e., the boundary of a convex body. Moreover, $\gamma^{\prime}$ pass through
the points $C,$ and $D$ when $\theta=\frac{5\pi}{4},$ and
$\theta=\frac{7\pi}{4},$ and share the tangent lines with $\Gamma$. This
ensure us that the body $K$, constructed as we said before, is a convex body.

Figure 9: $\gamma$ is a convex curve

We prove that the only convex body with two equipotential circles is the disc.

######  Theorem 2.

Let $K\subset\mathbb{R}^{2}$ be a convex body which possesses two
equipotential circles. Then $K$ is a disc.

_Proof._ By Theorem 1 we have that $K$ is centrally symmetric and the two
equipotential discs have centre at the centre of symmetry of $K$, to say $O$.
By Lemma 2 and Remark 1, we have that for at least one of the equipotential
discs it holds that the chords of $K$ and tangent to it, are seen from $O$
under a constant angle $\alpha\neq\frac{\pi}{2}$. By Theorem 1 we have that
$K$ is a disc. $\Box$

##  3 Some characterizations of the ellipsoid and the sphere

In this section we derive from Theorem 1 some characterizations of the sphere
and the ellipsoid.

######  Theorem 3.

Let $K\subset\mathbb{R}^{3}$ be a convex body and let $P,$ $Q$ be two points
in the interior of $K$. If every $2$-dimensional section of $K$ through $P$ or
$Q$ possesses an equipotential circle, then $K$ is an ellipsoid.

_Proof._ By Theorem 1 we have that every section of $K$ through $P$ or $Q$ has
a centre of symmetry. We recall the concept of a false centre: A convex body
$L$ has a false centre $X$ if every section of $L$ through $X$ has centre of
symmetry but $X$ is not centre of symmetry for $L$. Now we use the Theorem of
the False centre of Aitchison, Petty, Rogers, Montejano, and Morales (see [1],
and [12]) which asserts that a convex body with a false centre must be an
ellipsoid. Since either $P$ or $Q$ is not a centre of symmetry for $K$, we
have that one of them is a false centre and then $K$ is an ellipsoid. $\Box$

The following result is a particular case of a theorem due to S.
Olovjanishnikov [13], however, we give our own proof for the case we need in
some of the subsequent results. First we give some piece of notation: For
$\upsilon\in\mathbb{S}^{2}$, we denote by $\upsilon^{\perp}$ the subspace
orthogonal to $\upsilon$ and by $E(\upsilon)$ the affine plane
$\upsilon+\upsilon^{\perp}$. For $z\in\mathbb{R}^{3}\setminus\mathbb{B}$,
where $\mathbb{B}$ denote the unit ball of $\mathbb{R}^{3}$, we denote by
$C_{z}$ the union of all the supporting lines of $\mathbb{S}^{2}$ passing
through $z$.

######  Theorem 4.

Let $K\subset\mathbb{R}^{3}$ be a convex body. Suppose that
$\mathbb{S}^{2}\subset\emph{int}\,K$ and, for every
$\upsilon\in\mathbb{S}^{2}$, the section $E(\upsilon)\cap K$ is centrally
symmetric with centre at $\upsilon$. Then $K$ is a ball concentric with
$\mathbb{S}^{2}$.

_Proof._ First of all, we are going to prove that, for all $P\in\partial K$,
the projection of $\mathbb{S}^{2}$ from $P$ onto $\partial K$ is a circle. Let
$P\in\partial K$ and let $\Gamma$ be the circle $C_{P}\cap\mathbb{S}^{2}$ (see
Fig. 10). We will show that the set $\partial K\cap[C_{P}\setminus\\{P\\}]$,
which is denoted by $\Lambda$, is a circle. Let $W\in\Lambda$. Since the
segment $WP\subset C_{P}$, the line $WP$ makes contact with $\mathbb{S}^{2}$
in a point, say $Y$. It is clear that $Y\in\Gamma$. In virtue that $E(Y)\cap
K$ has centre at $Y$, $Y$ must be the midpoint of the chord $PW$. It follows
that $\Lambda=\phi(\Gamma)$, where
$\phi:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$ is an homothety with centre $P$
and ratio of homothety equal to 2. Therefore, $\Lambda$ is a circle and is
contained in a plane parallel to the plane of $\Gamma$.

Now we are going to prove that, for every $\upsilon\in\mathbb{S}^{2}$, the
section $E(\upsilon)\cap K$ has the following property: for every point
$X\in\partial[E(\upsilon)\cap K]$ there exist a supporting line $L(X)$ of
$E(\upsilon)\cap K$ such that $L(X)$ and the line $X\upsilon$ are orthogonal.
It is well known that such property characterizes the disc [18].

Consider an arbitrary point $\upsilon\in\mathbb{S}^{2}$ and a point
$X\in\partial[E(\upsilon)\cap K]$. We denote by $P$ the intersection of ray
$\overrightarrow{X\upsilon}$ with $\partial K$. By the first part of the
proof, there exist a plane $\Sigma$ such that $\partial
K\cap[C_{P}\setminus\\{P\\}]=\Sigma\cap\partial K$ is a circle, say again
$\Lambda$, and $X\in\Sigma\cap\partial K$. Since $E(\upsilon)$ is a supporting
plane of $\mathbb{S}^{2}$ (and of $C_{P}$), the line $E(\upsilon)\cap\Sigma$
is a supporting line of $\Lambda$. The line $E(\upsilon)\cap\Sigma$ must be
also a supporting line of $E(\upsilon)\cap K$, otherwise, there is a point
$Q\in E(\upsilon)\cap K$ such that $Q\in E(\upsilon)\cap\Sigma$ and $X\neq Q$.
Since $Q\in E(\upsilon)\cap\Sigma$ it follows that $Q\in\Sigma$. We have that
the points of $K$ which are in $\Sigma$ are those points in $\Lambda$, we
conclude that $Q\in\Lambda$, i.e., there is a point $Q$ in the supporting line
$E(\upsilon)\cap\Sigma$ of the circle $\Lambda$ different than $X$, which is
absurd. Hence the line $E(\upsilon)\cap\Sigma$ is a supporting line of
$E(\upsilon)\cap K$. Finally, we observe that, since $C_{P}$ is a right
circular cone, then $E(\upsilon)\cap\Sigma$ is orthogonal to the line
$X\upsilon$.

Figure 10: The section $E(\upsilon)\cap K$ is a circle

The proof of the theorem will be complete as soon as we prove the following.

Claim 3. If all the sections of a convex body $K\subset\mathbb{R}^{3}$, given
by supporting planes of a sphere $S\subset\text{int}K$, are circles and the
centre of each section is the contact point of the supporting plane with the
sphere, then $K$ is a ball concentric with $S$.

_Proof._ Suppose the centre and radius of $S$ are $O$ and $r$, respectively.
Let $H$ be an arbitrary plane tangent to $S$ at the point $A$ and denote by
$S_{A}$ the circle $H\cap K$. Let $K_{A}$ be the open region of $\partial K$
which is separated of $S$ by mean of the plane $H$. Consider any point $X\in
K_{A}$ and let $\Gamma_{X}$ be the any section of $K$ tangent to $S$ and
passing through $X$. Clearly, $\Gamma_{X}$ intersects the circle $S_{A}$ in
two points, to say $B,$ and $C$ and is tangent to $S$ in a point $D$. The
radius of $S_{A}$ and $S_{D}$ is the same and is equal to $|OB|^{2}-r^{2}$,
hence, $\overline{K_{A}}$ is a closed cap of a sphere with centre $O$ and
radius $|OB|$. Let $Z$ be any point in $S$ such that $K_{Z}\cap
K_{A}\neq\emptyset$. Again, $\overline{K_{Z}}$ is a closed cap of the sphere
with centre $O$ and radius $|OB|$. Continuing in this way, we obtain a closed
cover of $\partial K$, since $\partial K$ is a compact set, we have that there
exists a finite subcover of $\partial K$. Every region of this subcover is a
cap of the sphere with centre $O$ and radius $|OB|$, therefore, $K$ is a ball
concentric with $S$. $\Box$

The proof of Theorem 4 is now complete. $\Box$

As a corollary of Theorems 1 and 4 we have the following.

######  Theorem 5.

Let $K\subset\mathbb{R}^{3}$ be a convex body which contains a ball
$\mathcal{B}$ in its interior. Suppose every section of $K$, and tangent to
$\mathcal{B}$, has the contact point as the centre of an equipotential circle.
Then $K$ is a ball.

_Proof._ By Theorem 1 we have that every section of $K$ tangent to
$\mathcal{B}$ has the point of contact as centre of symmetry. By Theorem 4, we
have that $K$ and $\mathcal{B}$ are concentric balls, therefore, $K$ is a
ball. $\Box$

######  Remark 2.

If we consider any convex body $L$ in Theorem 5 instead of the ball
$\mathcal{B}$, the conclusion is that $K$ and $L$ are homothetic ellipsoids.
This conclusion follows if we apply Olovjanishnikov’s theorem (see [13]) which
asserts: _Let $K$ and $L$ be convex bodies in $\mathbb{R}^{n}$, with
$L\subset\text{int}\,K$, such that every hyper-section of $K$ tangent to $L$
is centrally symmetric and its centre belongs to $\partial L$, then $K$ and
$L$ are homothetic ellipsoids. _

A mixture between Theorems 3 and 5 is the following.

######  Theorem 6.

Let $K\subset\mathbb{R}^{3}$ be a convex body which contains a ball
$\mathcal{B}$ in its interior and let $P$ be a point in the interior of
$\mathcal{B}$, different from the centre of $\mathcal{B}$. Suppose every
section of $K$ through $P$ has as an equipotential circle, the intersection of
the section with $\partial\mathcal{B}$. Then $K$ is a ball.

_Proof._ First of all, we are going to prove that $P$ is a false centre of
$K$. By Theorem 1 we have that all sections of $K$ passing through $P$ are
centrally symmetric. We denote by $O$ the centre of $\mathcal{B}$. Since
$P\neq O$, there exist a line $L$ through $O$ such that $P$ does not belong to
$L$. Let $\Pi$ be the plane orthogonal to $L$ through $P$ and denote by $T$
the intersection $\Pi\cap L$. For one hand, since $P$ does not belong to $L$,
$P\neq T$, on the other hand, $T$ is the centre of the section
$\Pi\cap\mathcal{B}$ and, consequently, is the centre of $\Pi\cap K$. Hence
$P$ is a false centre of $K$. By the the False centre Theorem (see [1], and
[12]) $K$ is an ellipsoid. We denote the centre of $K$ by $R$. If $O\neq R$,
then the sections $\text{aff}\\{P,O,R\\}\cap\mathcal{B}$ and
$\text{aff}\\{P,O,R\\}\cap K$ (where $\text{aff}\\{P,O,R\\}$ denotes the plane
through $P,$ $O,$ and $R$), would have centres at $O$ and at $R$,
respectively. This would contradict Theorem 1 (if $P,O,R$ were collinear we
would consider any plane containing the line determined by this points).

On the other hand, is well known that the locus $\Omega_{E}(P)$ of the mid
points of the chords of the ellipsoid $E$, with centre at $O$, passing through
the point $P\in\text{int}E$ is an ellipsoid homothetic to $E$ and
$P,O\in\Omega_{E}(P)$. We denote by $\Delta_{E}(P)$ the locus of the centres
of the sections of $E$ passing through $P$. It is easy to see that
$\Omega_{E}(P)=\Delta_{E}(P)$. Thus $\Delta_{E}(P)$ is an ellipsoid homothetic
to $E$ and $P,O\in\Delta_{E}(P)$. By Theorem 1 it follows that
$\Delta_{K}(P)=\Delta_{S}(P)$, that is, $\Delta_{K}(P)$ is a sphere. Since
$\Delta_{K}(P)$ and $K$ are homothetic, we conclude that $K$ is a ball. $\Box$

For the next result we need the following definition: We say that a sphere
$\mathcal{S}$ in the interior of a convex body $K\subset\mathbb{R}^{3}$, is an
_equipotential sphere_ if for every point $P\in\mathcal{S}$, the tangent plane
at $P$ cuts a section $K(P)$ of $K$, such that $P$ is an equipotential point
of $K(P)$ with an equipotential constant $\lambda(P)$ (depending on the point
$P$).

######  Theorem 7.

Let $K\subset\mathbb{R}^{3}$ be a convex body which possesses and
equipotential sphere $\mathcal{S}$ with $\lambda(P)=\lambda_{0}$, for every
$P\in\mathcal{S}$. Then $K$ is a ball.

_Proof._ Let $P$ be any point in the interior of $\mathcal{S}$, different from
the centre of $S$. Consider any section $\mathcal{H}$ of $K$ through $P$,
since $\mathcal{S}$ is an equipotential sphere with an equipotential constant
independent of the point in $\mathcal{S}$, we have that $\mathcal{H}$ has the
circle $\mathcal{H}\cap\mathcal{S}$ as an equipotential circle. We have the
conditions of Theorem 6, therefore, $K$ must be a ball. $\Box$

##  4 Convex bodies with equireciprocal circles

Let $\mathcal{E}$ be an ellipse with foci $P$ and $Q$. Consider any chord $AB$
through $P$ (or $Q$), then the value of $\frac{1}{|AP|}+\frac{1}{|PB|}$ is
equal to a constant $\lambda$. If a point $P$ for a convex figure $K$ has the
property that for any chord $AB$ through $P$ the value of
$\frac{1}{|AP|}+\frac{1}{|PB|}$ is equal to a constant, we then say that $P$
is an _equireciprocal point_. In [4], K. Falconer proved that a $C^{2}$ convex
curve with two equireciprocal points must be an ellipse. However, if the
differentiability hypothesis is omitted, then there exist some convex curves,
different from ellipses, which have two equireciprocal points. In this section
we extent the notion of equireciprocal point: we say that a circle
$\mathcal{B}$ in the interior of a convex body $K\subset\mathbb{R}^{2}$ is an
_equireciprocal circle_ if for every chord $AB$ of $K$, tangent to
$\mathcal{B}$ at a point $P$ (depending on the chord $AB$), it holds that
$\frac{1}{|AP|}+\frac{1}{|PB|}=\lambda$, for some constant $\lambda$. It is
natural to ask about the existence of a convex body $K$, different from the
disc, which has an equireciprocal disc. The answer is given in the following.

######  Theorem 8.

Let $K\subset\mathbb{R}^{2}$ be a convex body containing an equireciprocal
disc $\mathcal{B}$ with centre $O$ in its interior. Then $K$ is a disc centred
at $O$.

_Proof._ The proof is completely analogous to the proof of Theorem 1 with only
one change: in Claim 2 we use the equation
$\cot\theta+\cot(\alpha-\theta)=\lambda$, instead of the equation
$\tan\theta\cdot\tan(\alpha-\theta)=\lambda.$ In this case, the derivative of
$f$ is

$f^{\prime}(\theta)=-\csc^{2}\theta+\csc^{2}(\alpha-\theta),$

which is negative for $\theta$ in the interval $(0,\alpha/2)$, independently
of the value of $\alpha\in(0,\pi)$. Hence, for the admissible value $\lambda$
there is only one solution of the mentioned equation. In other words, for
every value of $\alpha$, there is only one possible length for the chord $AB$.
Again, by the theorem of Barker and Larman [2], we conclude that $K$ is a disc
centred at $O$. $\Box$

We say that a sphere $\mathcal{S}$ in the interior of a convex body
$K\subset\mathbb{R}^{3}$, is an _equireciprocal sphere_ if for every point
$P\in\mathcal{S}$, the tangent plane at $P$ cuts a section $K(P)$ of $K$, such
that $P$ is an equireciprocal point of $K(P)$ with an equireciprocal constant
$\lambda(P)$ (depending on the point $P$). As a first corollary of Theorem 8
we have the following.

######  Theorem 9.

Let $K\subset\mathbb{R}^{3}$ be a convex body which contains an equireciprocal
sphere $\mathcal{S}$ in its interior, with $\lambda(P)=\lambda_{0}$ for every
$P\in\mathcal{S}$. Then $K$ is a ball.

_Proof._ Every section $\mathcal{H}$ of $K$ which intersects the interior of
$\mathcal{S}$, has $\mathcal{H}\cap\mathcal{S}$ as an equireciprocal circle,
hence $\mathcal{H}$ is a disc with centre at the centre of
$\mathcal{H}\cap\mathcal{S}$. Now, let $P$ be any point in the interior of
$\mathcal{S}$. We know that there exists an affine diameter of $K$ through $P$
(see for instance [17]), to say, $AB$. Every section $\mathcal{H}$ of $K$
which contains to $AB$, is a disc with centre at the midpoint of $AB$ and
radius equal to $|AB|/2$. It follows that $K$ is a ball with centre at the
midpoint of $AB$ and radius $|AB|/2$. $\Box$

For the following results we need a new definition (see [6]): Let
$\delta:\mathbb{S}^{2}\longrightarrow\mathbb{R}$ be a continuous function such
that $\delta(-x)=-\delta(x)$. We denote by $\Sigma$ the following set of
planes in $\mathbb{R}^{3}$:

$\Sigma=\\{H_{y}=\\{x\in\mathbb{R}^{3}|\langle
x,y\rangle=\delta(y)\\}\\}_{y\in\mathbb{S}^{2}},$

and we say that $\Sigma$ a $2$-cycle of planes. If for a given convex body
$K$, it holds that $H_{p}\cap H_{q}\cap\text{int}\,K\neq\emptyset$ for every
two planes $H_{p},H_{q}\in\Sigma$, we say that $\Sigma$ is a $2$-cycle of
planes for $K$. In [10], D. Larman, L. Montejano, and E. Morales proved some
results which characterizes the ellipsoid in terms of the properties of the
sections, cut by the members of a $2$-cycle of planes, in a given convex body.

We have the following result which is interesting by itself.

######  Theorem 10.

Let $K\subset\mathbb{R}^{3}$ a convex body and let $\Sigma$ a $2$-cycle of
planes for $K$. Suppose that for each $\Gamma\in\Sigma$ we have that
$\Gamma\cap K$ is a disc. Then $K$ is a ball.

_Proof._ Consider a unitary vector $u\in\mathbb{S}^{2}$, we may consider that
$\delta(u)$ is the distance with sign from $H_{u}$ to the origin $O$. Recall
that $H_{u}$ is the plane in $\Sigma$ that is orthogonal to $u$. Let $L$ be an
affine diameter of $K$ parallel to $u$.

Claim 4. There exists $\Pi\in\Sigma$ such that $L\subset\Pi$.

_Proof._ Let $\Sigma_{u}$ be the subset of $\Sigma$ such that for every
$H_{y}\in\Sigma_{u}$ it holds that $y$ is orthogonal to $u$. Denote by
$S(u)=\mathbb{S}^{2}\cap u^{\bot}.$ Let $\delta_{u}$ be the restriction of
$\delta$ to the set $S(u)$. Let $O^{\prime}$ be the midpoint of the affine
diameter $L$, and let $\sigma:S(u)\longrightarrow\mathbb{R}$ be the function
such that $\sigma(x)$ is the distance with sign from $O^{\prime}$ to $H_{x}$.
Since $\delta_{u}$ is a continuous function we have that $\sigma$ is also a
continuous function. Moreover, $\sigma(-x)=-\sigma(x)$, i.e., $\sigma$ is an
odd function. By the Borsuk-Ulam theorem (see for instance [11]), there exists
$x_{0}\in S(u)$ such that $\sigma(x_{0})=0$, this means that the plane
$H_{x_{0}}$ contains to $L$. $\Box$

We will show that there exists a plane of symmetry of $K$, orthogonal to $u$.
Let $\Delta$ be the plane orthogonal to $u$ that passes through the midpoint
of $L$. We assert that $\Delta$ is the plane of symmetry we are looking for.
For $X\in bd(K-\Pi)$, denote as $L_{X}$ the chord of $K$ through $X$ parallel
to $u$. Again, since $\Sigma$ is a $2$-cycle, there exists $\Pi_{X}\in\Sigma$
such that $L_{X}\subset\Pi_{X}$.

Since the line $\Pi\cap\Pi_{X}$ has inner points of $K$, then the discs
$\Pi\cap K$ and $\Pi_{X}\cap K$ have the common chord $(\Pi\cap\Pi_{X})\cap
K$. The plane $\Delta$ pass through the midpoint of the chord
$L_{X}\subset\Pi_{X}\cap K$, also, the centre of $\Pi_{X}\cap K$ is contained
in $\Delta$. Hence, the chord $\Delta\cap\Pi_{X}$ is a line of symmetry of the
disc $\Pi_{X}\cap K$, and the second point of intersection of $L_{X}$ with
$\Pi_{X}\cap K$, say $Y$, is at the same distance to the line
$\Delta\cap\Pi_{X}$ as $X$. It follows that $\Delta$ is a plane of symmetry of
$K$. We have proved that for every affine diameter $L$ of $K$, the plane
orthogonal to $L$ through its midpoint is a plane of orthogonal symmetry of
$K$, therefore, $K$ must be a ball. $\Box$

As a corollary of Theorems 8 and 10 we have our last result.

######  Corollary 1.

Let $K\subset\mathbb{R}^{3}$ a convex body and let $\Sigma$ a $2$-cycle of
planes. Suppose that for each $\Gamma\in\Sigma$ we have that $\Gamma\cap K$
possesses an equireciprocal circle. Then $K$ is a ball.

_Proof._ The proof is just an application of Theorems 8 and 10. $\Box$

##  5 Further comments and problems

In this section we give some problems and final comments.

Problem 1. Let $K\subset\mathbb{R}^{2}$ be a convex body which has a disc
$\mathcal{B}$ in its interior. Suppose every chord of $K$ which is tangent to
$\mathcal{B}$ subtends a constant angle $\alpha\neq\pi/2$ from the centre $O$
of $\mathcal{B}$. Is it $K$ a disc or an ellipse?

It is known that if $\alpha$ is a fixed angle, $\Omega$ an ellipse with foci
$A$ and $B$, then the envelope of the chords of $\Omega$, which are seen under
a constant angle $\alpha$ from a focus, to say $A$, is another ellipse with
one focus in $A$ (see [16]).

Figure 11: The envelope of the chords $CD$ is an ellipse

Problem 2. Let $K\subset\mathbb{R}^{3}$ be a convex body which has a ball
$\mathcal{B}$ in its interior. Suppose every section of $K$ which is tangent
to $\mathcal{B}$ subtends a solid constant angle $\alpha$ from the centre $O$
of $\mathcal{B}$. Is it $K$ a ball?

Problem 3. Is there an $O$-symmetric convex body $K\subset\mathbb{R}^{3}$,
different from a ball, which contains a ball $\mathcal{B}$ centred at $O$,
such that every section of $K$ through $O$ has as an equipotential circle, the
intersection of the section with $\partial\mathcal{B}$?

Problem 4. Let $K\subset\mathbb{R}^{3}$ be a convex body which possesses and
equipotential sphere $\mathcal{S}$, with the equipotential constant
$\lambda(P)$ not necessarily equal to a fixed number $\lambda_{0}$. Is it $K$
a ball?

Problem 5. Let $K\subset\mathbb{R}^{3}$ be a convex body which possesses and
equireciprocal sphere $\mathcal{S}$, with the equireciprocal constant
$\lambda(P)$ not necessarily equal to a fixed number $\lambda_{0}$. Is it $K$
a ball?

In a second paper on this topic, we will give more characterizations of the
sphere and ellipsoid in spaces of higher dimension. In these characterizations
we will use appropriate definitions for equipotential and equireciprocal
spheres.

##  6 Acknowledgment

We thank Diana Verdusco, Alejandro Estrada, Ulises Velasco, Carlos Yee, Luguis
de los Santos, and Sergei Tabachnikov, for many helpful discussions about the
topic of this work.

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