Normes sur Rn.ipynb

{
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  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "from IPython.display import HTML\n",
    "from ipywidgets import interact\n",
    "%matplotlib inline\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
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    "HTML('''<script>\n",
    "code_show=true; \n",
    "function code_toggle() {\n",
    " if (code_show){\n",
    " $('div.input').hide();\n",
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    "$( document ).ready(code_toggle);\n",
    "</script>\n",
    "The raw code for this IPython notebook is by default hidden for easier reading.\n",
    "To toggle on/off the raw code, click <a href=\"javascript:code_toggle()\">here</a>.''')"
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  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "<p><a href=\"https://commons.wikimedia.org/wiki/File:Cube_Watermelons_for_Sale_in_Munich,_Germany_Market.jpg#/media/File:Cube_Watermelons_for_Sale_in_Munich,_Germany_Market.jpg\"><img src=\"https://upload.wikimedia.org/wikipedia/commons/thumb/1/10/Cube_Watermelons_for_Sale_in_Munich%2C_Germany_Market.jpg/1200px-Cube_Watermelons_for_Sale_in_Munich%2C_Germany_Market.jpg\" alt=\"Cube Watermelons for Sale in Munich, Germany Market.jpg\" style=\"width: 50vw\"></a><br>By Francisco Antunes - <a rel=\"nofollow\" class=\"external free\" href=\"https://www.flickr.com/photos/francisco-antunes/3815031014/\">https://www.flickr.com/photos/francisco-antunes/3815031014/</a>, <a href=\"https://creativecommons.org/licenses/by/2.0\" title=\"Creative Commons Attribution 2.0\">CC BY 2.0</a>, <a href=\"https://commons.wikimedia.org/w/index.php?curid=70106531\">Link</a></p>\n"
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    "HTML(\"\"\"\n",
    "<p><a href=\"https://commons.wikimedia.org/wiki/File:Cube_Watermelons_for_Sale_in_Munich,_Germany_Market.jpg#/media/File:Cube_Watermelons_for_Sale_in_Munich,_Germany_Market.jpg\"><img src=\"https://upload.wikimedia.org/wikipedia/commons/thumb/1/10/Cube_Watermelons_for_Sale_in_Munich%2C_Germany_Market.jpg/1200px-Cube_Watermelons_for_Sale_in_Munich%2C_Germany_Market.jpg\" alt=\"Cube Watermelons for Sale in Munich, Germany Market.jpg\" style=\"width: 50vw\"></a><br>By Francisco Antunes - <a rel=\"nofollow\" class=\"external free\" href=\"https://www.flickr.com/photos/francisco-antunes/3815031014/\">https://www.flickr.com/photos/francisco-antunes/3815031014/</a>, <a href=\"https://creativecommons.org/licenses/by/2.0\" title=\"Creative Commons Attribution 2.0\">CC BY 2.0</a>, <a href=\"https://commons.wikimedia.org/w/index.php?curid=70106531\">Link</a></p>\n",
    "\"\"\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Des boules dans différentes normes du plan et de l'espace"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Pour un espace vectoriel normé donné $(E, ||.||)$, on définit la *boule unité* par $B(0,1) = \\{x \\in E, ||x|| <1 \\}.$ L'ensemble qu'on obtient dépend bien sûr de la norme sur $E$. \n",
    "\n",
    "Dans le cas du plan $\\mathbb{R}^2$, on utilise usuellement la *norme euclidienne* donnée par $$\\|(x_1,x_2)\\|_2 = \\sqrt{|x_1|^2+|x_2|^2}$$, et la boule qu'on obtient a effectivement une tête de boule:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "<video width=\"640\" controls>\n",
       "  <source src=\"Norme2.mp4\" type=\"video/mp4\">\n",
       "</video>\n"
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     "execution_count": 4,
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   "source": [
    "HTML(\"\"\"\n",
    "<video width=\"640\" controls>\n",
    "  <source src=\"Norme2.mp4\" type=\"video/mp4\">\n",
    "</video>\n",
    "\"\"\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On peut aussi munir $\\mathbb{R}^2$ de la *norme 1*:\n",
    "$$\\|(x_1,x_2)\\|_1 = |x_1|+|x_2|$$\n",
    "ou de la *norme infini*:\n",
    "$$\\|(x_1,x_2)\\|_\\infty = \\max\\{|x_1|^2+|x_2|^2\\}$$\n",
    "\n",
    "Les boules correspondantes n'ont pas la même tête !"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "<video width=\"640\" controls>\n",
       "  <source src=\"Norme1.mp4\" type=\"video/mp4\">\n",
       "</video>\n"
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       "<IPython.core.display.HTML object>"
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     "execution_count": 5,
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   "source": [
    "HTML(\"\"\"\n",
    "<video width=\"640\" controls>\n",
    "  <source src=\"Norme1.mp4\" type=\"video/mp4\">\n",
    "</video>\n",
    "\"\"\")"
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  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "<video width=\"640\" controls>\n",
       "  <source src=\"NormeInf.mp4\" type=\"video/mp4\">\n",
       "</video>\n"
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     "execution_count": 6,
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   "source": [
    "HTML(\"\"\"\n",
    "<video width=\"640\" controls>\n",
    "  <source src=\"NormeInf.mp4\" type=\"video/mp4\">\n",
    "</video>\n",
    "\"\"\")"
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  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Toutefois, ces normes ont quand même un lien entre elles: elles sont *équivalentes*.\n",
    "\n",
    "Deux normes $\\|.\\|_a$ et $\\|.\\|_b$ sur $E$ sont équivalentes s'il existe des constantes positives $c,\\ C$ telles que pour tout $x$, $c\\|x\\|_a \\leq \\|x\\|_b \\leq C\\|x\\|_a)$. L'équivalence des normes peut se réécrire en termes d'inclusion de boules: l'inégalité précédente se réécrit\n",
    "$$B_a(0,\\frac1C) \\subset B_b(0,1) \\subset B_a(0, \\frac1c) $$\n",
    "\n",
    "En l'occurrence, on a \n",
    "$$\\dfrac12 ||x||_1 \\leq ||x||_\\infty \\leq ||x||_2 \\leq \\sqrt{2} ||x||_\\infty\\leq \\sqrt{2} ||x||_1$$\n",
    "donc\n",
    "$$||x||_1 \\leq \\frac{1}{\\sqrt{2}} \\Rightarrow ||x||_\\infty\\leq \\frac{1}{\\sqrt{2}}\\Rightarrow ||x||_2 \\leq 1 \\Rightarrow ||x||_\\infty \\leq 1 \\Rightarrow ||x||_1 \\leq 2$$\n",
    "c'est-à-dire\n",
    "$$B_1(0,  \\frac{1}{\\sqrt{2}}) \\subset B_\\infty(0,  \\frac{1}{\\sqrt{2}})\\subset B_2(0,1) \\subset B_\\infty(0,1)\\subset B_1(0,2):$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "\n",
       "<video width=\"640\" controls>\n",
       "  <source src=\"inclusionBoules.mp4\" type=\"video/mp4\">\n",
       "</video>\n"
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     "execution_count": 7,
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   "source": [
    "HTML(\"\"\"\n",
    "<video width=\"640\" controls>\n",
    "  <source src=\"inclusionBoules.mp4\" type=\"video/mp4\">\n",
    "</video>\n",
    "\"\"\")"
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "En fait, pour tout $p \\geq 1$, on peut définir une norme sur $\\mathbb{R}^2$ par\n",
    "$$ \\|x\\|_p = (|x_1|^p + |x_2|^p)^{\\frac1p} $$\n",
    "Les boules unités associées passent continûment du losange de la norme 1, à la sphère usuelle, et tendent à se rapprocher du carré de la norme infinie quand $p$ devient grand:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
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      "text/plain": [
       "interactive(children=(FloatSlider(value=15.0, description='p', max=30.0, min=1.0, step=0.2), Output()), _dom_c…"
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    {
     "data": {
      "text/plain": [
       "<function __main__.update_plot(p)>"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import ipywidgets as widgets\n",
    "from IPython.display import display\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from math import pi\n",
    "\n",
    "\n",
    "\n",
    "p_min = 1   # the minimial value of the parameter a\n",
    "p_max = 30   # the maximal value of the parameter a\n",
    "p_init = 2   # the value of the parameter a to be used initially, when the graph is created\n",
    "\n",
    "x = np.linspace(-1, 1, 500)\n",
    "\n",
    "#fig,ax = plt.subplots(1,figsize=(6,6))\n",
    "\n",
    "\n",
    "\n",
    "def update_plot(p):\n",
    "    fig = plt.figure()\n",
    "    ax=fig.gca()\n",
    "    ax.clear()\n",
    "    plt.xlim(-1.5, 1.5)\n",
    "    plt.ylim(-1.5, 1.5)\n",
    "    plt.plot(x, (1-abs(x)**p)**(1/p), 'r')\n",
    "    plt.plot(x, -(1-abs(x)**p)**(1/p), 'r')\n",
    "    plt.show()\n",
    "\n",
    "\n",
    "#plt.title(\"Boule unité pour la norme p\")\n",
    "\n",
    "interact ( update_plot, p = (1,30,0.2) )\n",
    "\n",
    "#p=widgets.FloatSlider(min=p_min, max=p_max,value=p_init,description=\"p\")\n",
    "\n",
    "\n",
    "#widgets.interactive(update_plot, p=p)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Remarquons que quand $p$ est plus petit que 1, on n'obtient pas une norme, car l'inégalité triangulaire n'est plus vérifiée. Par exemple pour $p=\\frac12$, en prenant $e_1=(1,0)$, $e_2=(0,1)$, on a \n",
    "$$\\|e_1+e_2\\|_{\\frac12} = (\\sqrt{1}+\\sqrt{1})^2 = 4$$\n",
    "et\n",
    "$$\\|e_1\\|_\\frac12+\\|e_2\\|_\\frac12 =  (\\sqrt{0}+\\sqrt{1})^2+ (\\sqrt{1}+\\sqrt{0})^2 =2 $$\n",
    "donc $\\|e_1+e_2\\|_{\\frac12} > \\|e_1\\|_\\frac12+\\|e_2\\|_\\frac12$.\n",
    "Ceci se traduit par des boules concaves:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
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       "interactive(children=(FloatSlider(value=0.54, description='p', max=1.0, min=0.1, step=0.02), Output()), _dom_c…"
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     "data": {
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       "<function __main__.update_plot(p)>"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import ipywidgets as widgets\n",
    "from IPython.display import display\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from math import pi\n",
    "\n",
    "\n",
    "\n",
    "p_min = 0.1   # the minimial value of the parameter a\n",
    "p_max = 1  # the maximal value of the parameter a\n",
    "p_init = 0.5   # the value of the parameter a to be used initially, when the graph is created\n",
    "\n",
    "x = np.linspace(-1, 1, 500)\n",
    "\n",
    "#fig,ax = plt.subplots(1,figsize=(6,6))\n",
    "\n",
    "\n",
    "\n",
    "def update_plot(p):\n",
    "    fig = plt.figure()\n",
    "    ax=fig.gca()\n",
    "    ax.clear()\n",
    "    plt.xlim(-1.5, 1.5)\n",
    "    plt.ylim(-1.5, 1.5)\n",
    "    plt.plot(x, (1-abs(x)**p)**(1/p), 'r')\n",
    "    plt.plot(x, -(1-abs(x)**p)**(1/p), 'r')\n",
    "    plt.show()\n",
    "\n",
    "\n",
    "#plt.title(\"Boule unité pour la norme p\")\n",
    "\n",
    "interact ( update_plot, p = (0.1,1,0.02) )\n",
    "\n",
    "#p=widgets.FloatSlider(min=p_min, max=p_max,value=p_init,description=\"p\")\n",
    "\n",
    "\n",
    "#widgets.interactive(update_plot, p=p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Dans $\\mathbb{R}^3$, on peut de même définir, pour tout $p\\geq 1$, une *norme p* donnée par\n",
    "$$\\|x\\|_p =  (|x_1|^p + |x_2|^p + |x_3|^p)^{\\frac1p} $$.\n",
    "Les plus usuelles sont la norme 1, la norme 2 (ou euclidienne), et la norme infinie, qui donnent des boules comme ci-dessous:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
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       "<video width=\"640\" controls>\n",
       "  <source src=\"Norme1dim3.mp4\" type=\"video/mp4\">\n",
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       "<IPython.core.display.HTML object>"
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     "execution_count": 11,
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   "source": [
    "HTML(\"\"\"\n",
    "<video width=\"640\" controls>\n",
    "  <source src=\"Norme1dim3.mp4\" type=\"video/mp4\">\n",
    "</video>\n",
    "\"\"\")"
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   "execution_count": 12,
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       "  <source src=\"Norme2dim3.mp4\" type=\"video/mp4\">\n",
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     "execution_count": 12,
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   "source": [
    "HTML(\"\"\"\n",
    "<video width=\"640\" controls>\n",
    "  <source src=\"Norme2dim3.mp4\" type=\"video/mp4\">\n",
    "</video>\n",
    "\"\"\")"
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  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "<video width=\"640\" controls>\n",
       "  <source src=\"NormeInfdim3.mp4\" type=\"video/mp4\">\n",
       "</video>\n"
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     "execution_count": 13,
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   "source": [
    "HTML(\"\"\"\n",
    "<video width=\"640\" controls>\n",
    "  <source src=\"NormeInfdim3.mp4\" type=\"video/mp4\">\n",
    "</video>\n",
    "\"\"\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On peut, comme précédemment, faire varier $p$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
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    {
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      },
      "text/plain": [
       "interactive(children=(IntSlider(value=5, description='p', max=10, min=1), Output()), _dom_classes=('widget-int…"
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     },
     "metadata": {},
     "output_type": "display_data"
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    {
     "data": {
      "text/plain": [
       "<function __main__.update(p)>"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\n",
    "import matplotlib.pyplot as plt\n",
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "from matplotlib.widgets import Slider  # import the Slider widget\n",
    "import ipywidgets as widgets\n",
    "from IPython.display import display\n",
    "\n",
    "import numpy as np\n",
    "\n",
    "\n",
    "p_min = 1   # the minimial value of the parameter a\n",
    "p_max = 5   # the maximal value of the parameter a\n",
    "p_init = 2 \n",
    "\n",
    "\n",
    "n=10\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "def update(p):\n",
    "    fig = plt.figure()\n",
    "    ax = fig.gca(projection='3d')\n",
    "\n",
    "    ax.clear()\n",
    "    \n",
    "    x=[]\n",
    "    y=[]\n",
    "    for i in np.linspace(-1,1,2*n):\n",
    "        for j in np.linspace(-(1-np.abs(i)**p)**(1/p), (1-np.abs(i)**p)**(1/p),n):\n",
    "            x=x+[i]\n",
    "            y=y+[j]\n",
    "    x=np.array(x)\n",
    "    y=np.array(y)\n",
    "    z1=(np.abs(1-np.abs(x)**p-np.abs(y)**p))**(1/p)\n",
    "    z2=-z1\n",
    "    ax.plot_trisurf(x, y, z1, linewidth=0.2, antialiased=True, color='b')\n",
    "    ax.plot_trisurf(x, y, z2, linewidth=0.2, antialiased=True, color='b')\n",
    "    fig.canvas.draw_idle() \n",
    "    plt.show()\n",
    "\n",
    "#p=widgets.FloatSlider(min=p_min, max=p_max,value=p_init,description=\"p\")\n",
    "\n",
    "#widgets.interactive(update, p=p)\n",
    "\n",
    "interact ( update, p = (1,10,1) )\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "et on a, comme précédemment, l'équivalence des normes, traduite en inclusion de boules:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
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       "  <source src=\"Inclusion.mp4\" type=\"video/mp4\">\n",
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       "<IPython.core.display.HTML object>"
      ]
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     "execution_count": 15,
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   "source": [
    "HTML(\"\"\"\n",
    "<video width=\"640\" controls>\n",
    "  <source src=\"Inclusion.mp4\" type=\"video/mp4\">\n",
    "</video>\n",
    "\"\"\")"
   ]
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   "cell_type": "code",
   "execution_count": null,
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