On Convexity
upward and onward
Learning is rooted in repetition and convexity, meaning that the reading of a single text twice is more profitable than reading two different things once. - Nassim Nicholas Taleb
I was recently on a vacation with my wife in Mexico, whilst working out (tequila at night and weight training in the morning) I listened to a podcast with Nassim Nicholas Taleb - Mr. Black Swan himself. He was discussing the principle of convexity and why he believed it was one of the most important principles one should understand.
I agree with him, so in this essay I will cover convexity and how it can be applied to myriad domains. I will not cover the standard application of convexity (bonds), but rather how one can apply it to the areas of health, wealth, and wisdom.
In its simplest form convexity refers to the curvature or shape of a curve or a function. Convexity is a principle that can be applied to myriad aspects life.
Here are some commonly known areas convexity is present:
Optimization Theory: Convexity plays a central role in optimization problems. In mathematical optimization, a convex objective function or a convex set of constraints simplifies the search for the optimal solution because it guarantees that any local minimum found is also a global minimum.
Geometry: In geometry, convex polygons and polyhedra are shapes in which all interior angles are less than 180 degrees, and they have no "dents" or concave sections. Convex shapes are essential in various geometric algorithms and applications, such as computer graphics and computational geometry.
Physics: In physics, the concept of convexity can be applied to the shape of lenses and mirrors. Convex lenses, for example, are thicker at the center and thinner at the edges, causing them to converge light rays and focus them to a point.
Economics: Besides bond convexity, convexity also appears in economics when analyzing utility functions. A utility function is said to be convex if it exhibits a diminishing marginal rate of substitution, which means that as a consumer consumes more of one good, they are willing to give up fewer units of another good to maintain the same level of satisfaction.
Machine Learning and Statistics: Convex optimization is widely used in machine learning and statistics to train models and estimate parameters. Many optimization problems in these fields are convex, allowing for efficient and reliable optimization algorithms.
Transportation and Logistics: Convex optimization techniques are used in transportation and logistics to optimize routes, minimize costs, and allocate resources efficiently. This is particularly important in industries like supply chain management and transportation networks.
Materials Science: In materials science, the study of convexity is important in understanding the properties of materials, especially those related to their crystal structures and how atoms are arranged within materials.
Operations Research: Convex optimization techniques are applied in operations research to solve complex decision-making problems efficiently. For example, linear programming and quadratic programming are common convex optimization methods used in this field.
Chemistry: In chemistry, convexity concepts can be applied to the study of molecular structures and the shape of molecules, which can have significant implications for chemical reactivity and behavior.
Now onto some less intuitive applications of convexity as it pertains to health, wealth, and wisdom, but first, a story.
Julia
In the vibrant heart of Mexico City, a woman named Julia thrived amidst the bustling streets and rich culture of the metropolis. Julia was not your average city dweller; she possessed a deep curiosity about life and an unwavering commitment to improvement. Her quest for a well-rounded existence led her to embrace the principles of convexity in her pursuit of health, wealth, and philosophy.
Health:
Julia had always admired those who effortlessly maintained healthy lifestyles. Instead of jumping headfirst into a rigorous fitness regimen or extreme diet, she chose to apply convexity to her health. She began her journey with short, daily walks in the park, gradually increasing the distance and pace. Her diet transformed gradually as well, as she incorporated more fruits and vegetables while reducing processed foods. Julia realized that her newfound habits were sustainable and, over time, led to greater vitality.
Wealth:
Financial security was another realm of life where Julia sought to apply convexity. She didn't attempt to amass wealth overnight but rather adopted a habit of saving a small portion of her income regularly. As the years passed, she increased her contributions to savings and investments, understanding that the power of compounding would grow her wealth. Julia also paid off her debts systematically, starting with the smallest ones, and used the freed-up funds to tackle larger ones. Her financial acumen grew with each milestone achieved, and she remained patient, knowing that her efforts would bear fruit.
Wisdom:
Julia's interest in philosophy was her most profound pursuit. She realized that wisdom was not something one could gain overnight but rather a lifelong journey. She began with the basics, reading introductory philosophy texts and engaging in philosophical discussions with friends. Over time, she explored deeper and more complex topics, seeking the guidance of wise mentors. Julia knew that the true essence of philosophy lay in self-reflection and self-awareness. She incorporated mindfulness and meditation into her daily routine, gradually extending the duration of her practices. As her inner world expanded, so did her philosophical understanding of life.
As the years passed, Julia found herself thriving in all aspects of her life. Her health had improved, her wealth had grown steadily, and her philosophical insights had deepened. Her journey wasn't without setbacks or challenges, but Julia embraced them as part of her convex path to self-improvement.
One sunny afternoon, as Julia strolled through her favorite park, she paused to reflect on her journey. She had discovered that the principles of convexity were not just mathematical concepts but also guiding principles for a fulfilling life. With a sense of contentment and a smile on her face, she continued her walk, knowing that her journey of health, wealth, and wisdom was a lifelong adventure—one step at a time.
Wellness
Convexity can be applied to health and wellness in various ways, primarily in the context of optimization, decision-making, and behavior change. Here are some ways in which the concept of convexity can be applied:
Nutrition and Diet Planning: Convex optimization techniques can be used to optimize dietary choices. For example, given a set of nutritional goals (e.g., meeting daily vitamin and mineral requirements), a convex optimization model can find an optimal combination of foods that meet these goals while minimizing calorie intake or cost.
Exercise and Fitness: Convexity can be applied to design exercise routines that balance different types of physical activities (e.g., cardio, strength training, flexibility) to achieve fitness goals while considering time constraints and personal preferences.
Weight Management: In weight management programs, convex optimization can be used to develop personalized weight loss plans by considering an individual's dietary preferences, activity levels, and weight loss goals. The goal is to find an optimal plan that encourages sustainable weight loss.
Behavior Change: Convex optimization models can help design behavior change interventions by identifying the most effective and feasible strategies to encourage healthier habits. For example, they can optimize the allocation of resources to support smoking cessation or substance abuse treatment programs.
Mental Health and Well-Being: Convex optimization can be used to develop personalized mental health and well-being plans, considering factors like stress levels, coping mechanisms, and therapy options to maximize emotional well-being.
Chronic Disease Management: For individuals with chronic conditions, convex optimization can assist in developing personalized care plans that optimize medication schedules, doctor visits, and lifestyle modifications to improve overall health and manage the condition effectively.
In these applications, convexity helps optimize decisions and resource allocation in the context of health and wellness, making it a valuable tool for improving health outcomes and promoting overall well-being.
Wisdom
Applying convexity to wisdom is not a direct or common concept, as wisdom is a complex and multifaceted human trait that doesn't readily lend itself to mathematical or optimization frameworks. Wisdom encompasses aspects such as judgment, empathy, emotional regulation, and ethical considerations, which are deeply rooted in human experience and psychology. However, one could explore some indirect ways to conceptualize the relationship between convexity and wisdom:
Decision-Making: Wisdom often involves making decisions that balance various factors, including short-term and long-term consequences, ethical considerations, and the well-being of oneself and others. Convex optimization techniques could be applied to decision-making models that seek to find optimal solutions when facing complex choices, considering multiple objectives or constraints. However, these models would need to incorporate human values and ethical principles carefully.
Conflict Resolution: Wisdom often plays a role in resolving conflicts, whether in personal relationships, organizations, or international diplomacy. Convex optimization can be used to model and analyze conflict resolution strategies that seek to maximize mutual benefit and minimize harm in negotiations and agreements.
Moral Dilemmas: Wisdom often involves navigating moral and ethical dilemmas. Convexity could be applied to ethical decision-making frameworks, seeking solutions that balance different ethical principles and values. However, these frameworks would need to be based on comprehensive ethical theories and not oversimplified mathematical models.
Self-Reflection and Personal Growth: Wisdom often involves self-reflection and personal growth. While convexity may not directly apply here, data analytics and self-tracking tools could be used to help individuals gain insights into their behavior, emotions, and decision-making processes, potentially contributing to personal development and the accumulation of wisdom over time.
Educational and Training Programs: Convexity could be applied to optimize educational and training programs that aim to develop wisdom-related skills such as empathy, emotional intelligence, and conflict resolution. These programs could use data and optimization techniques to tailor interventions to individual learners' needs.
It's essential to note that wisdom is a deeply human and subjective concept that is not easily quantifiable or reducible to mathematical models. While mathematical and computational techniques can be valuable tools for understanding and improving various aspects of human life, they must be used with caution when dealing with abstract and complex qualities like wisdom. Wisdom is often best nurtured through life experiences, reflection, and learning from one's mistakes and successes.
Optimization
Convexity can be indirectly applied to self-improvement in the context of setting and achieving personal goals, making decisions, and managing habits. Here are some ways in which convexity principles can be used for self-improvement:
Goal Setting and Planning: When setting self-improvement goals, it's helpful to break them down into smaller, manageable tasks or milestones. This is known as the "convexification" of goals. Instead of aiming for a daunting long-term objective, dividing it into smaller, achievable sub-goals can make the journey more manageable and less overwhelming. Achieving these smaller milestones can provide a sense of accomplishment and motivation to continue progressing.
Habit Formation: Convexity can be applied to building positive habits and breaking negative ones. Rather than attempting radical and unsustainable changes, gradual adjustments to daily routines can lead to better long-term outcomes. This aligns with the concept of "convex fitness functions," where small improvements are rewarded, making it more likely to maintain positive habits over time.
Time Management: Convexity principles can be used in time management by allocating time to tasks and activities based on their importance and urgency. Applying the Eisenhower Matrix, which categorizes tasks into four quadrants (urgent and important, not urgent but important, urgent but not important, neither urgent nor important), can help prioritize and focus on important but not necessarily urgent tasks, fostering personal growth and development.
Learning and Skill Development: When acquiring new skills or knowledge, a convex approach involves consistent practice or learning sessions over time rather than cramming or intensive, short-term efforts. This spaced repetition and gradual accumulation of expertise can be more effective for long-term retention and skill development.
Health and Fitness: Convexity can be applied to fitness goals and health improvement. Gradual increases in exercise intensity or dietary changes can help avoid burnout or setbacks associated with extreme changes. The goal is to create a sustainable fitness routine or healthy eating habits that align with long-term well-being.
Personal Finance: Convexity principles can be used in personal finance by gradually increasing savings and investments over time. Starting small and consistently contributing to savings or retirement accounts can lead to significant financial growth due to the compounding effect. This is why retirement accounts are the best option for most people.
Personal Development Plans: When designing a personal development plan, consider using convex optimization techniques to prioritize and schedule activities that align with your goals. This approach can help you make efficient use of your time and resources to achieve personal growth.
Feedback and Reflection: Regular self-assessment and reflection are essential for self-improvement. Convexity can be applied to feedback loops, where small adjustments are made based on continuous self-assessment, leading to incremental improvements.
Emotional Intelligence and Self-Awareness: Convexity can be indirectly applied by recognizing that personal growth is not linear. It's essential to accept setbacks and failures as part of the learning process, allowing for resilience and adaptability in the face of challenges.
Applying convexity to self-improvement involves adopting a gradual, incremental, and sustainable approach to personal growth and development. By breaking down goals, managing time effectively, and cultivating positive habits over time, individuals can increase their chances of achieving lasting improvements in various aspects of their lives.
Furthermore, applying the principles of convexity to health, wealth, and philosophy involves adopting a gradual and sustainable approach to these areas of life. Here's how you can apply convexity principles to each domain:
Health:
Fitness and Exercise: Instead of pushing yourself to the limit with intense workouts, adopt a gradual approach. Start with manageable exercise routines and progressively increase intensity and duration over time. This allows your body to adapt and reduces the risk of burnout or injury.
Diet and Nutrition: Gradually introduce healthier food choices and portion control into your diet. Avoid extreme diets and focus on long-term sustainable changes. Small, consistent adjustments in your eating habits can lead to significant improvements in health.
Mental Health: Prioritize self-care and stress management. Regularly practice relaxation techniques, mindfulness, or meditation. These practices, when done consistently, can improve mental well-being and emotional resilience.
Sleep: Build better sleep habits by gradually adjusting your bedtime and wake-up times. Avoid drastic changes to your sleep schedule, as this can disrupt your circadian rhythm. Quality sleep is crucial for overall health.
Wealth:
Savings and Investments: Apply the principle of regular contributions and compounding to your financial strategy. Start with a small savings or investment plan and consistently increase your contributions over time. This approach leverages the power of compounding to grow your wealth gradually.
Debt Reduction: If you have debts, prioritize paying them off gradually. Target the debts with the highest interest rates first. Allocate a portion of your income toward debt repayment while still meeting your basic expenses. As you pay off one debt, roll the payment into paying off the next one, accelerating your progress.
Financial Goals: Set achievable financial goals and break them down into smaller milestones. As you achieve these milestones, your financial confidence and competence will grow, motivating you to tackle more significant financial challenges.
Crypto
Leveraging the principles of convexity in the cryptocurrency realm can help individuals and investors manage risks and optimize their strategies in this highly volatile and speculative market. Here are some ways to apply convexity principles in cryptocurrency:
Diversification: Diversify your cryptocurrency portfolio across different assets and tokens. By spreading your investments, you reduce the risk associated with the extreme volatility of individual cryptocurrencies. Diversification follows convexity principles as it aims to minimize potential losses while maximizing the potential for gains.
Risk Management: Apply risk management strategies, such as setting stop-loss orders and defining acceptable levels of risk. These measures help protect your capital and align with convexity by limiting potential downside while allowing for potential upside.
Dollar-Cost Averaging (DCA): Implement a DCA strategy by investing a fixed amount of money at regular intervals, regardless of the current cryptocurrency price. This strategy is convex because it ensures that you buy more when prices are low and less when prices are high, potentially reducing the impact of market volatility on your overall investment.
Position Sizing: Use a position sizing strategy that adapts your investment size based on the volatility and risk associated with each cryptocurrency. Allocate a smaller percentage of your portfolio to highly volatile assets, reducing the impact of potential losses.
Staking and Yield Farming: Engage in staking and yield farming activities to earn passive income from your cryptocurrency holdings. These activities can provide a steady stream of rewards, aligning with convexity principles by generating income regardless of market fluctuations.
Hedging Strategies: Consider using options or other hedging instruments to protect your cryptocurrency holdings from adverse price movements. Hedging strategies aim to limit losses while allowing for potential gains.
Long-Term Holding: Adopt a long-term investment mindset and hold cryptocurrencies with strong fundamentals that you believe in for the long haul. This strategy is convex because it minimizes the impact of short-term price fluctuations on your overall portfolio, allowing for potential gains over time.
Research and Due Diligence: Apply a diligent research process before investing in any cryptocurrency. Understand the technology, team, use case, and market potential of each asset. This approach follows convexity by minimizing the risk of investing in speculative or poorly-researched projects.
Continuous Learning: Stay informed about the cryptocurrency market and emerging trends. Continuously adapt your investment strategy based on new information and market developments, maintaining a flexible approach in line with convexity principles.
Emotional Discipline: Control emotional responses to market volatility. Avoid impulsive buying or selling decisions driven by fear or greed. Staying emotionally disciplined aligns with convexity by reducing the likelihood of making hasty decisions that may result in significant losses.
Cryptocurrency markets are known for their extreme volatility, and leveraging the principles of convexity can help individuals navigate this environment with a more balanced and risk-aware approach. It's essential to remember that while convexity-based strategies can mitigate risks, they do not eliminate them entirely, and investing in cryptocurrencies remains inherently speculative and carries risks. Therefore, careful consideration and a long-term perspective are crucial when applying these principles in the crypto realm.
Philosophy:
Ethical Decision-Making: Apply the principles of ethical reasoning gradually. Study ethical theories and practice ethical decision-making in everyday life. Reflect on your actions and decisions, seeking to align them with your evolving ethical framework.
Philosophical Inquiry: Dive into philosophical topics one step at a time. Start with foundational philosophical ideas and gradually explore more complex concepts. Engage in critical thinking and thoughtful discussions with others to expand your philosophical perspective.
Wisdom and Self-Reflection: Cultivate wisdom through self-reflection and introspection. Accept that wisdom is a lifelong journey and that setbacks and mistakes are part of the learning process. Continually seek to improve your understanding of the world and your role within it.
Mindfulness and Meditation: Incorporate mindfulness and meditation practices into your daily routine. Gradually build your meditation practice, starting with short sessions and increasing duration as you become more comfortable. These practices can deepen your philosophical exploration by enhancing self-awareness and insight.
Applying convexity principles to health, wealth, and philosophy involves recognizing the value of gradual progress and sustainable change. Avoiding extreme or abrupt measures in these areas can lead to more balanced, healthier, and wiser outcomes over time. Remember that personal growth and development are ongoing processes, and patience and consistency are key to achieving lasting results.
Back to Julia
In the vibrant heart of Mexico City, Julia's journey was a testament to the transformative power of embracing the principles of convexity. Her life unfolded as a harmonious blend of health, wealth, and philosophy, and her approach was one of patience, gradual progress, and unwavering commitment.
Julia's pursuit of health was characterized by the wisdom to take small, sustainable steps. She walked the path of fitness and nutrition with grace, recognizing that lasting vitality came from gradual transformations rather than drastic changes. Her journey was a testament to the enduring value of balanced health practices.
In the realm of wealth, Julia demonstrated that financial security could be achieved through steady, disciplined efforts. She saved and invested patiently, understanding that wealth accumulation was a long-term endeavor. By systematically paying off her debts, she freed herself from financial burdens and cultivated financial wisdom.
Julia's philosophical journey was perhaps the most profound of all. She embraced the philosophy of gradual self-reflection and self-awareness, delving into the depths of wisdom one thought at a time. Her exploration of complex ideas and her meditation practices deepened her understanding of life's profound questions.
As the years passed, Julia's life blossomed in all aspects. Her health thrived, her wealth grew steadily, and her philosophical insights deepened. Her story was a testament to the beauty of a life well-lived, one that embodied the principles of convexity—small, consistent steps leading to enduring transformation.
Julia's journey was a reminder that the pursuit of health, wealth, and philosophy is not a sprint but a lifelong adventure. By following the path of gradual progress and embracing the wisdom of patience, she found fulfillment and contentment in her vibrant world, continuing her journey—one step at a time.
In summary, convexity is a versatile concept that appears in a wide range of disciplines, and its application often involves the analysis of shapes, optimization, and decision-making in various contexts.